Skip to main content

Full text of "Rocket propulsion elements"

See other formats


ROCKET 

PROPULSION 

ELEMENTS 

Eighth Edition 




GEORGE P. SUTTON 
OSCAR BIBLARZ 




ROCKET PROPULSION ELEMENTS 



The book’s cover shows a typical separation of a payload stage or satellite from 
an upper stage of a space flight vehicle in the vacuum of space. There are two 
rocket propulsion systems. The liquid propellant rocket engine on the upper stage 
(on the lower left) has just lifted the satellite into space and has just been shut 
down. The reaction control system on the payload stage has four small inclined 
thrusters, which can be seen during their firing operation; they are moving the 
payload away from the expended upper stage. A small thruster’s plume (flame) 
has a bright, elliptically shaped, hot gas, very small core, which is visible. The 
hot exhaust gases disperse from this core in all directions, and this part of the 
plume is almost invisible. 



Rocket Propulsion Elements 



Eighth Edition 



GEORGE P. SUTTON 

Consultant 

Formerly Laboratory Associate 

Lawrence Livermore National Laboratory and formerly 

Executive Director, Engineering 

Rocketdyne Division of The Boeing Company 

Now Pratt & Whitney Rocketdyne 

OSCAR BIBLARZ 

Professor Emeritus 

Department of Mechanicai and Astronauticai Engineering 
Naval Postgraduate School 




WILEY 

JOHN WILEY & SONS, INC. 




This book is printed on acid-free paper.® 

Copyright © 2010 hy John Wiley & Sons, Inc. All rights reserved 
Published by John Wiley & Sons, Inc., Hohoken, New Jersey 
Published simultaneously in Canada 

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any 
form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, 
except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, with- 
out either the prior written permission of the Publisher, or authorization through payment of the 
appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 
01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to 
the Publisher for permission should be addressed to the Permissions Department, John Wiley & 
Sons, Inc., Ill River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at 
www.wiley.com/go/permissions. 

Limit of Liability/Disclaimer of Warranty: While the publisher and the authors have used their 
best efforts in preparing this book, they make no representations or warranties with respect to the 
accuracy or completeness of the contents of this book and specihcally disclaim any implied warranties 
of merchantability or fitness for a particular purpose. No warranty may be created or extended by 
sales representatives or written sales materials. The advice and strategies contained herein may not 
be suitable for your situation. You should consult with a professional where appropriate. Neither the 
publisher nor the authors shall be liable for any accidents or loss of proht or any other commercial 
damages, including but not limited to special, incidental, consequential, or other damages. 

For general information about our other products and services, please contact our Customer Care 
Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 
or fax (317) 572-4002. 

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print 
may not be available in electronic books. For more information about Wiley products, visit our web 
site at www.wiley.com. 

Library of Congress Cataloging-in-Publication Data: 

Sutton, George Paul. 

Rocket propulsion elements / George P. Sutton, Oscar Biblarz. — 8th ed. 
p. cm. 

Includes bibliographical references and index. 

ISBN 978-0-470-08024-5 (cloth) 

1. Rocket engines. I. Biblarz, Oscar. II. Title. 

TL782.S8 2010 
629.47'5— dc22 

2009042591 



Printed in the United States of America 



10 987654321 



CONTENTS 



PREFACE xiii 

1 Classification 1 

1.1. Duct Jet Propulsion / 2 

1 .2. Rocket Propulsion / 4 

1.3. Applications of Rocket Propulsion / 14 
References / 26 

2 Definitions and Fundamentals 28 

2.1. Definitions / 28 

2.2. Thrust / 33 

2.3. Exhaust Velocity / 36 

2.4. Energy and Efficiencies / 38 

2.5. Multiple Propulsion Systems / 40 

2.6. Typical Performance Values / 41 
Problems / 43 

Symbols / 45 
References / 46 

3 Nozzle Theory and Thermodynamic Relations 47 



3.1. Ideal Rocket / 48 

3.2. Summary of Thermodynamic Relations / 49 



V 



VI 



CONTENTS 



3.3. Isentropic Flow Through Nozzles / 53 

3.4. Nozzle Configurations / 75 

3.5. Real Nozzles / 85 

3.6. Nozzle Alignment / 96 
Problems / 97 
Symbols / 100 
References / 101 

4 Flight Performance 103 

4.1. Gravity-Free Drag-Free Space Flight / 103 

4.2. Forces Acting on a Vehicle in the Atmosphere / 107 

4.3. Basic Relations of Motion / 110 

4.4. Space Flight / 116 

4.5. Flight Maneuvers / 131 

4.6. Effect of Propulsion System on Vehicle Performance / 137 

4.7. Flight Vehicles / 139 

4.8. Military Missiles / 148 

4.9. Flight Stability / 152 
Problems / 153 
Symbols / 155 
References / 157 

5 Chemical Rocket Propellant Performance Analysis 158 

5.1. Background and Fundamentals / 159 

5.2. Analysis of Chamber or Motor Case Conditions / 165 

5.3. Analysis of Nozzle Expansion Processes / 170 

5.4. Computer-Assisted Analysis / 175 

5.5. Results of Thermochemical Calculations / 176 
Problems / 190 

Symbols / 191 
References / 192 

6 Liquid Propellant Rocket Engine Fundamentals 194 

6.1. Types of Propellants / 198 

6.2. Propellant Tanks / 201 

6.3. Propellant Feed Systems / 208 

6.4. Gas Pressure Feed Systems / 210 

6.5. Tank Pressurization / 217 

6.6. Turbopump Feed Systems and Engine Cycles / 221 



CONTENTS Vii 

6.7. Rocket Engines for Maneuvering, Orbit Adjustments, or Attitude 

Control / 229 

6.8. Engine Eamilies / 233 

6.9. Valves and Pipelines / 236 

6.10. Engine Support Structure / 239 
Problems / 240 

Symbols / 242 
References / 243 

7 Liquid Propellants 245 

7.1. Propellant Properties / 246 

7.2. Liquid Oxidizers / 256 

7.3. Liquid Euels / 259 

7.4. Liquid Monopropellants / 263 

7.5. Gelled Propellants / 265 

7.6. Gaseous Propellants / 266 

7.7. Safety and Environmental Concerns / 267 
Problems / 268 

Symbols / 269 
References / 269 

8 Thrust Chambers 271 

8.1. Injectors / 276 

8.2. Combustion Chamber and Nozzle / 285 

8.3. Low-Thrust Rocket Thrust Chambers or Thrusters / 301 

8.4. Materials and Eabrication / 305 

8.5. Heat Transfer Analysis / 311 

8.6. Starting and Ignition / 323 

8.7. Random Variable Thrust / 326 

8.8. Sample Thrust Chamber Design Analysis / 328 
Problems / 339 

Symbols / 342 
References / 344 

9 Liquid Propellant Combustion and Its Stability 346 

9.1. Combustion Process / 346 

9.2. Analysis and Simulation / 350 

9.3. Combustion Instability / 352 
Problems / 364 
References / 364 



CONTENTS 



10 Turbopumps and Their Gas Supplies 366 

10.1. Introduction / 366 

10.2. Descriptions of Several Turbopumps / 367 

10.3. Selection of Turbopump Configuration / 372 

10.4. Flow, Shaft Speeds, Power, and Pressure Balances / 376 

10.5. Pumps / 378 

10.6. Turbines / 387 

10.7. Approach to Turbopump Preliminary Design / 391 

10.8. Gas Generators and Preburners / 394 
Problems / 395 

Symbols / 396 
References / 397 

1 1 Engine Systems, Controls, and Integration 399 

11.1. Propellant Budget / 399 

11.2. Performance of Complete or Multiple Rocket Propulsion 

Systems / 401 

11.3. Engine Design / 403 

11.4. Engine Controls / 411 

11.5. Engine System Calibration / 423 

11.6. System Integration and Engine Optimization / 430 
Problems / 432 
Symbols / 432 
References / 433 

Propellant Rocket Fundamentals 435 

Basic Relations and Propellant Burning Rate / 437 
Other Performance Issues / 458 
Propellant Grain and Grain Configuration / 463 
Propellant Grain Stress and Strain / 472 

Attitude Control and Side Maneuvers with Solid Propellant Rocket 
Motors / 483 
Problems / 485 
Symbols / 488 
References / 489 

13 Solid Propellants 492 

13.1. Classihcation / 492 

13.2. Propellant Characteristics / 498 



12 Solid 

12 . 1 . 

12 . 2 . 

12.3. 

12.4. 

12.5. 



CONTENTS 



IX 



13.3. Hazards / 505 

13.4. Propellant Ingredients / 512 

13.5. Other Propellant Categories / 522 

13.6. Liners, Insulators, and Inhibitors / 526 

13.7. Propellant Processing and Manufacture / 529 
Problems / 532 

References / 535 

14 Solid Propellant Combustion and Its Stability 537 

14.1. Physical and Chemical Processes / 537 

14.2. Ignition Process / 541 

14.3. Extinction or Thrust Termination / 543 

14.4. Combustion Instability / 545 
Problems / 553 
References / 554 

15 Solid Rocket Components and Motor Design 556 

15.1. Motor Case / 556 

15.2. Nozzles / 564 

15.3. Igniter Hardware / 578 

15.4. Rocket Motor Design Approach / 583 
Problems / 590 

References / 592 

16 Hybrid Propellant Rockets 594 

16.1. Applications and Propellants / 596 

16.2. Interior Hybrid Motor Ballistics / 600 

16.3. Performance Analysis and Grain Configuration / 604 

16.4. Design Example / 609 

16.5. Combustion Instability / 613 
Problems / 617 

Symbols / 618 
References / 620 

17 Electric Propulsion 622 

17.1. Ideal Plight Performance / 628 

17.2. Electrothermal Thrusters / 633 

17.3. Nonthermal Electrical Thrusters / 640 

17.4. Optimum Flight Performance / 656 



X 



CONTENTS 



17.5. Mission Applications / 661 

17.6. Electric Space-Power Supplies and Power-Conditioning 

Systems / 663 
Problems / 667 
Symbols / 669 
References / 671 

18 Thrust Vector Control 

18.1. TVC Mechanisms with a Single Nozzle / 675 

18.2. TVC with Multiple Thrust Chambers or Nozzles / 685 

18.3. Testing / 687 

18.4. Integration with Vehicle / 687 
Problems / 688 
References / 689 

19 Selection of Rocket Propulsion Systems 

19.1. Selection Process / 693 

19.2. Criteria for Selection / 699 

19.3. Interfaces / 705 
References / 709 

20 Rocket Exhaust Plumes 

20.1. Plume Appearance and Flow Behavior / 712 

20.2. Plume Effects / 725 

20.3. Analysis and Mathematical Simulation / 729 
Problems / 730 

References / 731 

21 Rocket Testing 

21.1. Types of Tests / 733 

21.2. Test Facilities and Safeguards / 735 

21.3. Instrumentation and Data Management / 742 

21.4. Flight Testing / 746 

21.5. Postaccident Procedures / 747 
References / 748 



673 



691 



710 



733 



CONTENTS 



XI 



Appendix 1 Conversion Factors and Constants 749 

Appendix 2 Properties of the Earth’s Standard Atmosphere 752 

Appendix 3 Summary of Key Equations for Ideal Chemical 

Rockets 753 

Index 755 



PREFACE 



This eighth edition follows earlier editions and focuses on the subject of rocket 
propulsion, its fundamentals, its essential technologies, and its key design ratio- 
nale. It describes the various types of rocket propulsion systems, physical phe- 
nomena, and essential relationships. It aims to balance a rigorous introduction of 
fundamentals with tables, figures, and recent information that can be of interest 
to practitioners. It endeavors to provide a clearer explanation of the basic physics 
and chemistry of rocket propulsion and some of its design approaches. 

Several chapters are devoted exclusively to liquid propellant rocket engines 
and also to solid propellant rocket motors (almost half the book), the two most 
prevalent of the rocket propulsion systems. It also has one chapter each on hybrid 
propulsion (with one liquid and one solid propellant) and on electrical propul- 
sion (with several different types). The principal flight applications for these 
four major types of rocket propulsion systems are described. Brief discussions 
of nuclear thermal propulsion systems and solar thermal propulsion system are 
included. Our book does not contain speculative future propulsion schemes such 
as nuclear fusion propulsion or pulsed detonations of propellant charges, because 
these are as yet not considered to be practical. 

The first few chapters of the book concern basic subjects. Following an intro- 
duction are definitions of the key propulsion parameters, principal thermodynamic 
relations, nozzle theory, vehicle flight analyses (flying inside and outside the 
earth’s atmosphere), or thermochemical analyses of combustion gas properties. 
The last few chapters relate intimately to rocket propulsion systems, namely thrust 
vector control, selection of propulsion systems for specific missions, exhaust 
plumes, and the testing of rocket propulsion systems. 

The book has been used for six decades in undergraduate and graduate courses 
of rocket propulsion in many universities, both in the United States and in other 

xiii 



XIV 



PREFACE 



countries. In this eighth edition many examples and problems have been revised 
or added for a better discussion of the principles involved and the technical issues 
used in preliminary design. More than 20 new homework problems have been 
added (and several others were deleted) for more relevance of the text material 
and for more practice of the key equations. Some of the problems have printed 
answers, so students can self-check their solutions. Many of the problems are 
patterned after the analyses and engineering efforts of actual propulsion systems. 
A solutions manual is available to selected users. Most of the book may be 
covered in a one- or two-semester course depending on the students’ background. 

Since the first edition in 1949 it has been a most popular technical rocket 
propulsion book and has been used by at least 75,000 students and professionals 
in approximately 37 countries. According to a committee of the American Insti- 
tute of Aeronautics and Astronautics it is the longest living aerospace book ever, 
in print continuously for 60 years. It is the only aerospace book cited in two 
prestigious book awards of the American Institute of Aeronautics and Astronau- 
tics. Earlier editions have been translated into Russian, Chinese, and Japanese. 
The authors have given guest lectures on rocket propulsion subjects at several 
universities and a three-day course based on this book. 

A number of new items and a few extensively changed sections appear in this 
new eighth edition. For example, this is the first edition with a separate chapter 
on turbopumps, an essential component of medium and large liquid propellant 
rocket engines. It has new or extensively revised discussions on tank pressur- 
ization, a new class of solid propellants that have been put into production, the 
first high-thrust large hybrid propulsion system that has flown in an experimental 
manned craft with nitrous oxide as the oxidizer and a rubbery fuel, a concept 
for integrated computer programs for modern rocket propulsion engines, health 
monitoring controls able to save hardware, and new flight applications for elec- 
trical propulsion systems. Almost all libraries of rocket propulsion organizations 
seem to have at least two editions of this book. The new edition has about three 
dozen new and different figures, a dozen revised or new tables, and many other 
clarifications, deletions, changes, or minor additions. 

It is usually easier, faster, and less expensive to uprate or modify an existing 
proven propulsion system for a new application than to design and build a truly 
new one. For example, in the liquid propellant engine field with turbopump feed 
systems, it has been more common in the last few decades to increase the thrust or 
performance by modifying an existing engine than to undertake the development 
of a brand new engine. A revised discussion of engine families (with related 
engine designs) and a new table of 13 different models of the RF 10 upper-stage 
engine emphasized this point. 

Although the United States is officially committed to convert from the English 
Engineering (EE) system of units (foot, pound) to the SI or metric system of units 
(m, kg), this has not as yet happened. The major portion of the U.S. propulsion 
industry for chemical rockets has been and is currently still using EE units for 
design and manufacturing. If a particular customer wants SI units, they will be 
given engineering information in SI units, but the fabrication will most likely 



PREFACE 



XV 



Still be done in EE units. Many colleges and research organizations in the United 
States and most propulsion organizations in the other countries use the SI system 
of units. So this book uses both systems; the examples and problems use one or 
the other and some tables have both sets of units. 

Computers have changed the way we do business in many fields. In rocket 
propulsion computer programs have been developed for many engineering anal- 
yses, computer-aided designs, computer-aided manufacturing, business and engi- 
neering transactions, test data collection, data analysis, data presentation, project 
management, and many others. In fact computers are used extensively in some 
companies to design new propulsion systems. Therefore we identify in this book 
the places where computer programs will be helpful and we mentions this often. 
However, we do not discuss specific computer programs because they take up too 
much space, become obsolete in a short time without regular upgrading, some 
do not have a way to provide help to a user, and some of the better programs 
are company proprietary and thus not publicly available. 

Propellants are hazardous materials. The authors and the publisher recom- 
mend that readers of this book do not work with them or handle them without 
exhaustive study of the hazards, the behavior, and properties of each propellant, 
and rigorous safety training including becoming familiar with protective equip- 
ment. Safety training and propellant information is given routinely to employees 
of organizations in this business. Neither the authors nor the publisher assume 
any responsibility for actions on rocket propulsion taken by the readers, either 
directly or indirectly. The information presented in this book is insufficient and 
inadequate for conducting rocket propulsion experiments or operations. 

The authors acknowledge gratefully the data, suggestions, and assistance given 
to them while preparing the manuscript for this book. Fortunately we have had 
the help from a number of outstanding experts in the field, reviewers, and edi- 
tors. We appreciate the reviews of solid propellant motors by David McGrath 
of ATK Launch Systems, Inc., and reviews of several chapters and the supply 
of certain propellant data by Eckart W. Schmidt, formerly of Olin Aerospace 
and currently a consultant in hazardous materials, is appreciated. Information on 
electrical propulsion was obtained from Scott Benson, of NASA Glen Research 
Center. Christopher Brophy of the Naval Post-Graduate School in Monterey made 
suggestions on hybrid propulsion. Thanks to Thomas L. Moore formerly with 
CPIAC (Chemical Propulsion Information Analysis Center) and now with ATK 
Systems for a variety of information, such as the production of a new category of 
solid propellant. Thanks also to Nick Heim of CPIAC for data on various motors, 
permissible exposure limits for toxic vapors, and gelled propellants. Frederick S. 
Simmons of The Aerospace Corporation reviewed the chapter on exhaust plumes 
and provided additional data on plumes. James H. Morehart of The Aerospace 
Corporation furnished precise detailed data on several rocket propulsion systems. 
Aerojet General Corporation of Sacramento (they elected to remain anonymous 
and did not single out specific deserving employees) provided data on a large 
solid rocket booster. Jeffrey S. Kincaid of Pratt & Whitney Rocketdyne (PWR) 
of Canoga Park, California, supplied the concept of integrated computer systems 



XVI 



PREFACE 



and corrections to engine data, and Dennis Ramine of PWR in Canoga Park sent 
information on heat transfer. Vince J. Wheelock formerly of Rocketdyne fur- 
nished several key figures. Patricia Mills of PWR at West Palm Beach, Florida, 
collected and sent data on the RL 10 engine family. Dan Magers of John Wiley 
and Sons, Inc. updated references and the editors and printers of our publisher 
made a coarse manuscript into a useful book. 

George P. Sutton 
Los Angeles, California 

Oscar Biblarz, 

Monterey, California 



CHAPTER 1 



CLASSIFICATION 



The word propulsion comes from the Latin propulsus, which is the past participle 
of the verb propellere, meaning to drive away. In a broad sense propulsion is the 
act of changing the motion of a body. Propulsion mechanisms provide a force 
that moves bodies that are initially at rest, changes a velocity, or overcomes 
retarding forces when a body is propelled through a medium. Jet propulsion is 
a means of locomotion whereby a reaction force is imparted to a device by the 
momentum of ejected matter. 

Rocket propulsion is a class of jet propulsion that produces thrust by ejecting 
matter stored in a flying vehicle called the propellant. Duct propulsion is a class 
of jet propulsion and includes turbojets and ramjets; these engines are also com- 
monly called air-breathing engines. Duct propulsion devices utilize mostly the 
surrounding medium as the “working fluid,” together with some vehicle-stored 
fuel. Combinations of rockets and duct propulsion devices can be attractive for 
some narrow applications and one is briefly described in this chapter. 

The energy source most useful to rocket propulsion is chemical combustion. 
Energy can also be supplied by solar radiation and, in the past, also by nuclear 
reaction. Accordingly, the various propulsion devices can be divided into chem- 
ical propulsion, nuclear propulsion, and solar propulsion. Table 1-1 lists many 
of the important propulsion concepts according to their energy source and type 
of propellant or working fluid. Radiation energy can originate from sources other 
than the sun and theoretically can cover the transmission of energy by microwave 
and laser beams, electromagnetic waves, and electrons, protons, and other par- 
ticle beams from a transmitter to a flying receiver. Nuclear energy is associated 
with the transformations of atomic particles within the nucleus of atoms and 
usually is created by fission or fusion. Other energy sources, both internal (in the 
vehicle) and external, can be considered. The energy form found in the output 



1 



2 CLASSIFICATION 



TABLE 1-1. Energy Sources and Propellants for Various Propulsion Concepts 



Propulsion Device 


Energy Source" 
Chemical Nuclear 


Solar 


Propellant or Working Fluid 


Turbojet 


D/P 


TFD 




Fuel -F air 


Turbo-ramjet 


TFD 






Fuel -F air 


Ramjet (hydrocarbon fuel) 


D/P 


TFD 




Fuel -F air 


Ramjet (H2 cooled) 


TFD 






Hydrogen -F air 


Rocket (chemical) 


D/P 


TFD 




Stored propellant 


Ducted rocket 


TFD 






Stored solid fuel -F 










surrounding air 


Electric rocket 


D/P 


TFD 


D/P 


Stored propellant 


Nuclear fission rocket 




TFD 




Stored H2 


Nuclear fusion rocket 




TFND 




Stored H2 


Solar-heated rocket 






TFD 


Stored H2 


Photon rocket (hig light 




TFND 




Photon ejection (no stored 


bulb) 








propellant) 


Solar sail 






TFD 


Photon reflection (no stored 










propellant) 



“D/P, developed and/or considered practical; TFD, technical feasibility has been demonstrated, but 
development is incomplete; TFND, technical feasibility has not yet been demonstrated. 



of a rocket is largely the kinetic energy of the ejected matter; thus the rocket 
converts the input from the energy source into this form. The ejected mass can 
be in a solid, liquid, or gaseous state. Often a combination of two or more of 
these is ejected. At very high temperatures it can also be a plasma, which is an 
electrically activated gas. 



1.1. DUCT JET PROPULSION 

This class, also called air-breathing engines, comprises devices which have a duct 
to confine the flow of air. They use oxygen from the air to burn fuel stored in 
the flight vehicle. The class includes turbojets, turbofans, ramjets, and pulsejets. 
This class of propulsion is mentioned primarily to provide a comparison with 
rocket propulsion and a background for combination rocket-duct engines, which 
are mentioned later. Table 1-2 compares several performance characteristics 
of specific chemical rockets with those of typical turbojets and ramjets. A high 
specific impulse, which is a measure of performance to be defined later, is directly 
related to a long flight range and thus indicates the superior range capability of air 
breather engines over chemical rockets at relatively low altitude. The uniqueness 
of the rocket, for example, high thrust to weight, high thrust to frontal area, and 
nearly thrust independence of altitude, enables extremely long flight ranges to be 
obtained in rarefied air and in space. 

The turbojet engine is the most common of ducted engines. Figure 1-1 shows 
the basic elements. 



1.1. DUCT JET PROPULSION 3 



TABLE 1-2. Comparison of Several Characteristics of a Typical Chemical Rocket and 
Two-Duct Propulsion Systems 



Feature 


Chemical 
Rocket Engine 
or Rocket 
Motor 


Turbojet Engine 


Ramjet Engine 


Thrust-to-weight ratio, 


75:1 


5:1, turbojet and 


7:1 at Mach 3 at 


typical 




afterburner 


30,000 ft 


Specific fuel consumption 


8-14 


0.5-1.5 


2.3-3.5 


(pounds of propellant or 
fuel per hour per pound 
of thrust)® 


Specific thrust (pounds of 


5000-25,000 


2500 (low Mach at 


2700 (Mach 2 at sea 


thrust per square foot 




sea level) 


level) 


frontal area)'’ 


Thrust change with altitude 


Slight increase 


Decreases 


Decreases 


Thrust vs. flight speed 


Nearly constant 


Increases with 


Increases with speed 


Thrust vs. air temperature 


Constant 


speed 

Decreases with 


Decreases with 


Flight speed vs. exhaust 


Unrelated, 


temperature 
Flight speed 


temperature 
Flight speed always 


velocity 


flight speed 


always less than 


less than exhaust 




can be 


exhaust velocity 


velocity 


Altitude limitation 


greater 

None; suited to 


14,000-17,000 m 


20,000 m at Mach 3 


Specific impulse, typical® 


space travel 
270 sec 


1600 sec 


30.000 m at Mach 5 

45.000 m at Mach 12 
1400 sec 


(thrust force per unit 
propellant or fuel weight 
flow per second) 



“Multiply by 0.102 to convert to kg/(hr-N). 

'’Multiply by 47.9 to convert to N/m^ 

‘^Specific impulse is a performance parameter and is defined in Chapter 2. 



At supersonic flight speeds above Mach 2, the ramjet engine (a pure duct 
engine) becomes attractive for flight within the atmosphere. Thrust is produced 
by increasing the momentum of the air as it passes through the ramjet, basically as 
is accomplished in the turbojet and turbofan engines but without compressors or 
turbines. Figure 1-2 shows the basic components of one type of ramjet. Ramjets 
with subsonic combustion and hydrocarbon fuel have an upper speed limit of 
approximately Mach 5; hydrogen fuel, with hydrogen cooling, raises this to at 
least Mach 16. Ramjets with supersonic combustion are known as scramjets and 
have flown in experimental vehicles. All ramjets depend on rocket boosters or 
some other method (such as being launched from an aircraft) for being accelerated 



4 CLASSIFICATION 




FIGURE 1-1. Simplified schematic diagram of a turbojet engine. 



Fuel 




FIGURE 1-2. Simplified diagram of a ramjet with a supersonic inlet (converging and 
diverging flow passage). 



to near their design flight speed to become functional. The primary applications of 
ramjets with subsonic combustion have been in shipboard and ground-launched 
antiaircraft missiles. Studies of a hydrogen-fueled ramjet for hypersonic aircraft 
look promising. The supersonic flight vehicle is a combination of a ramjet-driven 
high-speed airplane and a one- or two-stage rocket booster. It can travel at speeds 
up to a Mach number of 25 at altitudes of up to 50,000 m. 



1.2. ROCKET PROPULSION 

Rocket propulsion systems can be classified according to the type of energy 
source (chemical, nuclear, or solar), the basic function (booster stage, sustainer 
or upper stages, attitude control, orbit station keeping, etc.), the type of vehicle 
(aircraft, missile, assisted takeoff, space vehicle, etc.), size, type of propellant, 
type of construction, or number of rocket propulsion units used in a given vehicle. 

Another way is to classify by the method of producing thrust. A thermody- 
namic expansion of a gas is used in the majority of practical rocket propulsion 
concepts. The internal energy of the gas is converted into the kinetic energy of 
the exhaust flow and the thrust is produced by the gas pressure on the surfaces 
exposed to the gas, as will be explained later. This same thermodynamic the- 
ory and the same generic equipment (nozzle) is used for jet propulsion, rocket 
propulsion, nuclear propulsion, laser propulsion, solar-thermal propulsion, and 
some types of electrical propulsion. Totally different methods of producing thrust 



1.2. ROCKET PROPULSION 5 



are used in other types of electric propulsion or by using a pendulum in a gravity 
gradient. As described below, these electric systems use magnetic and/or electric 
fields to accelerate electrically charged molecules or atoms at very low densities. 
It is also possible to obtain a very small acceleration by taking advantage of the 
difference in gravitational attraction as a function of altitude, but this method is 
not explained in this book. 

The Chinese developed and used solid propellant in rocket missiles over 800 
years ago, and military bombardment rockets were used frequently in the eigh- 
teenth and nineteenth centuries. However, the significant developments of rocket 
propulsion took place in the twentieth century. Early pioneers included the Rus- 
sian Konstantin E. Ziolkowsky, who is credited with the fundamental rocket 
flight equation and his 1903 proposals to build rocket vehicles. The German 
Hermann Oberth developed a more detailed mathematical theory; he proposed 
multistage vehicles for space flight and fuel-cooled thrust chambers. The Amer- 
ican Robert H. Goddard is credited with the first flight using a liquid propellant 
rocket engine in 1926. An early book on the subject was written by the Viennese 
engineer Eugen Sanger. For rocket history see Refs. 1-1 to 1-8. 

Chemical Rocket Propulsion 

The energy from a high-pressure combustion reaction of propellant chemicals, 
usually a fuel and an oxidizing chemical, permits the heating of reaction product 
gases to very high temperatures (2500 to 4100°C or 4500 to 7400°F). These 
gases subsequently are expanded in a nozzle and accelerated to high velocities 
(1800 to 4300 m/sec or 5900 to 14,100 ft/sec). Since these gas temperatures are 
about twice the melting point of steel, it is necessary to cool or insulate all the 
surfaces that are exposed to the hot gases. According to the physical state of 
the propellant, there are several different classes of chemical rocket propulsion 
devices. 

Liquid propellant rocket engines use liquid propellants that are fed under pres- 
sure from tanks into a thrust chamber.* A typical pressure-fed liquid propellant 
rocket engine system is schematically shown in Fig. 1-3. The liquid bipropellant 
consists of a liquid oxidizer (e.g., liquid oxygen) and a liquid fuel (e.g., kerosene). 
A monopropellant is a single liquid that contains both oxidizing and fuel species; 
it decomposes into hot gas when properly catalyzed. A large turbopump-fed liq- 
uid propellant rocket engine is shown in Fig. 1-4. Gas pressure feed systems are 
used mostly on low thrust, low total energy propulsion systems, such as those 
used for attitude control of flying vehicles, often with more than one thrust cham- 
ber per engine. Pump-fed liquid rocket systems are used typically in applications 
with larger amounts of propellants and higher thrusts, such as in space launch 
vehicles. See Refs. 1-1 to 1-6. 

*The term thrust chamber, used for the assembly of the injector, nozzle, and chamber, is preferred 
by several official agencies and therefore has been used in this book. For small spacecraft control 
rockets the term thruster (a small thrust chamber) is commonly used, and this term will be used in 
some sections of this book. 



6 CLASSIFICATION 



Check Check 

valve valve 




FIGURE 1-3. Schematic flow diagram of a liquid propellant rocket engine with a gas 
pressure feed system. The dashed lines show a second thrust chamber, but some engines 
have more than a dozen thrust chambers supplied by the same feed system. Also shown 
are components needed for start and stop, controlling tank pressure, filling propellants 
and pressurizing gas, draining or flushing out remaining propellants, tank pressure relief 
or venting, and several sensors. 



In the thrust chamber the propellants react to form hot gases, which in turn are 
accelerated and ejected at a high velocity through a supersonic nozzle, thereby 
imparting momentum to the vehicle. A nozzle has a converging section, a con- 
striction or throat, and a conical or bell-shaped diverging section as further 
described in the next two chapters. 

Some liquid rocket engines permit repetitive operation and can be started and 
shut off at will. If the thrust chamber is provided with adequate cooling capacity, 
it is possible to run liquid rockets for periods exceeding 1 hour, dependent only on 



1.2. ROCKET PROPULSION 7 




FIGURE 1-4. Simplified schematic diagram of one type of liquid propellant rocket 
engine with a turbopump feed system and a separate gas generator, which generates 
“warm” gas for driving the turbine. Not shown are components necessary for controlling 
the operation, filling, venting, draining, or flushing out propellants, filters or sensors. This 
turbopump assembly consists of two propellant pumps, a gear case, and a high speed 
turbine. 



the propellant supply. A liquid rocket propulsion system requires several precision 
valves and a complex feed mechanism which includes propellant pumps, turbines, 
or a propellant-pressurizing device, and a relatively intricate combustion or thrust 
chamber. 

In solid propellant rocket motors* the propellant to be burned is contained 
within the combustion chamber or case (see Fig. 1-5). The solid propellant 



‘Historically, the word engine is used for a liquid propellant rocket propulsion system and the word 
motor is used for solid propellant rocket propulsion. They were developed originally by different 
groups. 



8 CLASSIFICATION 



Nozzle throat 

insert Nozzle exit cone 




FIGURE 1-5. Simplified perspective three-quarter section of a typical solid propellant 
rocket motor with the propellant grain bonded to the case and the insulation layer and 
with a conical exhaust nozzle. The cylindrical case with its forward and aft hemispherical 
domes form a pressure vessel to contain the combustion chamber pressure. Adapted with 
permission from Ref. 12-1. 



charge is called the grain and it contains all the chemical elements for complete 
burning. Once ignited, it usually bums smoothly at a predetermined rate on all the 
exposed internal surfaces of the grain. Initial burning takes place at the internal 
surfaces of the cylinder perforation and the four slots. The internal cavity grows 
as propellant is burned and consumed. The resulting hot gas flows through the 
supersonic nozzle to impart thrust. Once ignited, the motor combustion proceeds 
in an orderly manner until essentially all the propellant has been consumed. There 
are no feed systems or valves. See Refs. 1-7 to 1-10. 

Liquid and solid propellants, and the propulsion systems that use them, are dis- 
cussed in Chapters 6 to 11 and 12 to 15, respectively. Liquid and solid propellant 
rocket propulsion systems are compared in Chapter 19. 

Gaseous propellant rocket engines use a stored high-pressure gas, such as air, 
nitrogen, or helium, as their working fluid or propellant. The stored gas requires 
relatively heavy tanks. These cold gas engines have been used on many early 
space vehicles for low thmst maneuvers and for attitude control systems and 
some are still used today. Heating the gas by electrical energy or by combus- 
tion of certain monopropellants improves the performance and this has often 
been called warm gas propellant rocket propulsion . Chapter 7 discusses gaseous 
propellants. 

Hybrid propellant rocket propulsion systems use both a liquid and a solid 
propellant. For example, if a liquid oxidizing agent is injected into a combus- 
tion chamber filled with a solid carbonaceous fuel grain, the chemical reaction 
produces hot combustion gases (see Fig. 1-6). They are described further in 
Chapter 16. Several have flown successfully. 



1.2. ROCKET PROPULSION 9 




FIGURE 1-6. Simplified schematic diagram of a typical hybrid rocket engine. The 
relative positions of the oxidizer tank, high-pressure gas tank, and the fuel chamber with 
its nozzle depend on the particular vehicle design. 



Combinations of Ducted Jet Engines and Rocket Engines 

The Tomahawk surface-to-surface missile uses two stages of propulsion in 
sequence. The solid propellant rocket booster lifts the missile away from its 
launch platform and is discarded after its operation. A small turbojet engine 
sustains the low-level flight at nearly constant speed toward the target. 

A ducted rocket, sometimes called an air-augmented rocket, combines the 
principles of rocket and ramjet engines; it gives higher performance (specific 
impulse) than a chemical rocket engine, while operating within the earth’s atmo- 
sphere. Usually the term air-augmented rocket denotes mixing of air with the 
rocket exhaust (fuel rich for afterburning) in proportions that enable the propul- 
sion device to retain the characteristics typifying a rocket engine, for example, 
high static thrust and high thrust-to-weight ratio. In contrast, the ducted rocket 
often is like a ramjet in that it must be boosted to operating speed and uses the 
rocket components more as a fuel-rich gas generator (liquid or solid). 

The principles of the rocket and ramjet can be combined. An example of 
these two propulsion systems operating in sequence and in tandem and yet 
utilize a common combustion chamber volume as shown in Fig. 1-7. The low- 
volume configuration, known as an integral rocket-ramjet , can be attractive in 
air-launched missiles using ramjet propulsion. The transition from the rocket 
to the ramjet requires enlarging the exhaust nozzle throat (usually by ejecting 
rocket nozzle parts), opening the ramjet air inlet-combustion chamber interface, 
and following these two events with the normal ramjet starting sequence. 

A solid fuel ramjet uses a grain of solid fuel that gasifies or ablates and 
reacts with air. Good combustion efficiencies have been achieved with a patented 
boron-containing solid fuel fabricated into a grain similar to a solid propellant 
and burning in a manner similar to a hybrid rocket propulsion system. 



10 CLASSIFICATION 



Solid rocket 




Fuel manifold-. propellant \ 



Ejectable 



^ rocket 
^nozzle 
inserts 



^ ^Ramjet 
nozzle 



FIGURE 1-7. Elements of an air-launched missile with integral rocket-ramjet propul- 
sion. After the solid propellant has been consumed in boosting the vehicle to flight speed, 
the rocket combustion chamber becomes the ramjet combustion chamber with air burning 
the ramjet liquid fuel. 

Nuclear Rocket Engines 

Two different types of nuclear energy sources have been investigated for deliv- 
ering heat to a working fluid, usually liquid hydrogen, which subsequently can 
be expanded in a nozzle and thus accelerated to high ejection velocities (6000 
to 11,000 m/sec). However, none can be considered fully developed today and 
none have flown. They are ihe, fission reactor and the fusion reactor. Both are 
basically extensions of liquid propellant rocket engines. The heating of the gas is 
accomplished by energy derived from transformations within the nuclei of atoms. 
In chemical rockets the energy is obtained from within the propellants, but in 
nuclear rockets the power source is usually separate from the propellant. 

In the nuclear fission reactor rocket, heat can be generated by the fission of 
uranium in the solid reactor material and subsequently transferred to the work- 
ing fluid (see Refs. 1-11 to 1-13). The nuclear fission rocket is primarily a 
high- thrust engine (above 40,000 N) with specific impulse values up to 900 sec. 
Fission rockets were designed and tested in the 1960s in the United States and 
also in the Soviet Union, which today is Russia. Ground tests with hydrogen 
as a working fluid culminated in a thrust of 980,000 N (210,0001b force) at a 
graphite core nuclear reactor power level of 4100 MW with an equivalent altitude- 
specific impulse of 848 sec and a hydrogen temperature of about 2500 K. There 
were concerns with the endurance of the materials at the high temperature (above 
2600 K) and intense radiations, power level control, cooling a reactor after oper- 
ation, moderating the high-energy neutrons, and designing lightweight radiation 
shields for a manned space vehicle. No further ground tests of nuclear fission 
rocket engines have been undertaken. 

In recent years there has been renewed interest in nuclear fission rocket propul- 
sion primarily for a potential manned planetary exploration mission. Studies 
have shown that the high specific impulse (estimated in some studies at 1100 
sec) allows shorter interplanetary trip transfer times, smaller vehicles, and more 
flexibility in the launch time when planets are not in their optimum relative 
position. See Refs. 1-11 and 1-13. 



1.2. ROCKET PROPULSION 



11 



Fusion is an alternate way to create nuclear energy, which can heat a working 
fluid. A number of different concepts have been studied. To date none are feasible 
or practical. 

Concerns about an accident with the inadvertent spreading of radioactive mate- 
rials in the earth environment and the high cost of development programs have 
to date prevented a renewed experimental development of a large nuclear rocket 
engine. Unless there are some new findings and a change in world attitude about 
nuclear radiation, it is unlikely that a nuclear rocket engine will be developed or 
flown in the next few decades. Therefore no further discussion of it is given in 
this book. 



Electric Rocket Propulsion 

In all electric propulsion the source of the electric power (nuclear, solar radiation 
receivers, or batteries) is physically separate from the mechanism that produces 
the thrust. This type of propulsion requires a heavy and inefficient power sources 
as discussed below. The thrust usually is low, typically 0.005 to 1 N. In order to 
allow a significant increase in the vehicle velocity, it is necessary to apply the 
low thrust and thus a small acceleration for a long time (weeks or months) (see 
Chapter 17 and Refs. 1-14 and 1-15). 

Of the three basic types, electrothermal rocket propulsion most resembles 
the previously mentioned chemical rocket units; propellant is heated electrically 
(by heated resistors or electric arcs), and the hot gas is then thermodynamically 
expanded and accelerated to supersonic velocity through an exhaust nozzle (see 
Fig. 1-8). These electrothermal units typically have thrust ranges of 0.01 to 0.5 N, 
with exhaust velocities of 1000 to 5000 m/sec, and ammonium, hydrogen, nitro- 
gen, or hydrazine decomposition product gases have been used as propellants. 

The two other types — the electrostatic or ion propulsion engine and the 
electromagnetic or magnetoplasma engine — accomplish propulsion by differ- 
ent principles, and the thermodynamic expansion of gas in a nozzle, as such, 
does not apply. Both will work only in a vacuum. In an ion rocket (see Fig. 1-9) 
a working fluid (typically, xenon) is ionized (by stripping off electrons), and then 
the electrically charged heavy ions are accelerated to very high velocities (2000 
to 60,000 m/sec) by means of electrostatic fields. The ions are subsequently elec- 
trically neutralized; they are combined with electrons to prevent the buildup of 
a space charge on the vehicle. 

In the magnetoplasma rocket an electrical plasma (an energized hot gas con- 
taining ions, electrons, and neutral particles) is accelerated by the interaction 
between electric currents and magnetic fields and ejected at high velocity (1000 
to 50,000 m/sec). There are many different types and geometries. The Hall-effect 
thruster, which accelerates a plasma, has a good flight record in Russia. A simple 
pulsed (not continuously operating) electrical propulsion unit with a solid pro- 
pellant is shown in Fig. 1-10. It has had a good flight record as a spacecraft 
attitude control engine. 



12 CLASSIFICATION 




Arc between cathode 
tip and annular 
region of anode 



FIGURE 1-8. Simplified schematic diagram of arc-heating electric rocket propulsion 
system. The arc plasma temperature is very high (perhaps 15,000 K) and the anode, 
cathode, and chamber will get hot (1000 K) due to heat transfer. 




FIGURE 1-9. Simplified schematic diagram of a typical ion rocket, showing the approx- 
imate distribution of the electric power. 

Other Rocket Propulsion Concepts 

Several teehnologies exist for harnessing solar energy to provide the power for 
spacecraft and also to propel spacecraft using electrical propulsion. Solar cells 
generate electric power from the sun’s radiation. They are well developed and 
have been successful for several decades. Most electric propulsion systems have 
used solar cells for their power supply. Batteries and isotope decay power sources 
have also been used. 

One concept is the solar thermal racket', it has large-diameter optics to concen- 
trate the sun’s radiation (e.g., by lightweight precise parabolic mirrors or Fresnel 
lenses) onto a receiver or optical cavity. Figure 1-11 shows one concept and 
some data is given in Table 2-1. The receiver is made of high-temperature metal 
(such as tungsten or rhenium) and has a cooling jacket or heat exchanger. It 
heats a working fluid, usually liquid hydrogen, up to perhaps 2500°C and the hot 
gas is controlled by hot gas valves and exhausted through one or more nozzles. 
The large mirror has to be pointed toward the sun, and this usually requires the 
mirror to be adjustable in its orientation as the spacecraft orbits around the earth. 



1.2. ROCKET PROPULSION 13 




Plasma exhaust from 
parallel rail nozzle 



FIGURE 1-10. Simplified diagram of a rail accelerator for self-induced magnetic accel- 
eration of a current-carrying plasma. When the capacitor is discharged, an arc is struck at 
the left side of the rails. The high current in the plasma arc induces a magnetic field. The 
action of the current and the magnetic field causes the plasma to be accelerated at right 
angles to both the magnetic held and the current, namely in the direction of the rails. 
Each time the arc is created a small amount of solid propellant (Teflon) is vaporized and 
converted to a small plasma cloud, which (when ejected) gives a small pulse of thrust. 
Actual units can operate with many pulses per second. 



Heat receiver 




Radiation 
from the 
sun 



FIGURE 1-11. Simplified schematic diagram of a solar thermal rocket concept. 



Performance can be two to three times higher than that of a chemical rocket and 
thrust levels in most studies are low (1 to 10 N). Since large lightweight optical 
elements cannot withstand drag forces without deformation, the optical systems 
are deployed outside the atmosphere. Contamination is negigible, but storage or 
refueling of liquid hydrogen is a challenge. Problems being investigated include 




14 CLASSIFICATION 



rigid, lightweight mirror or lens structures, operational life, minimizing hydrogen 
evaporation, and heat losses to other spacecraft components. To date the solar 
thermal rocket has not yet provided the principal thrust of a flying spacecraft. 

The solar sail is another concept. It is basically a big photon reflector surface. 
The power source for the solar sail is the sun and it is external to the vehicle 
(see Ref. 1-16). Approaches using nuclear explosions and pulsed nuclear fusion 
have been analyzed (Refs. 1-17 and 1-18) but are not yet feasible. Concepts 
for transmitting radiation energy (by lasers or microwaves) from ground stations 
on Earth to satellites have been proposed but are not yet developed. 

International Rocket Propulsion Effort 

Active development or production of rocket propulsion systems was or is under 
way in more than 30 different countries. Some of them have made signihcant 
and original contributions to the state of the art of the technologies. There is 
mention in this book of a few foreign rocket units and their accomplishments 
and references to international rocket literature. Although most of the data in 
this book are taken from U.S. rocket experience, this is not intended to minimize 
foreign achievements. 

At the time of this writing the major international program has been the 
International Space Station (ISS), a multiyear cooperative effort with major con- 
tributions from the United States and Russia and active participation by several 
other nations. This manned orbital space station is used for conducting experi- 
ments and observations on a number of research projects. See Ref. 1-19. 



1.3. APPLICATIONS OF ROCKET PROPULSION 

Because the rocket can reach a performance unequaled by other prime movers, 
it has its own fields of application and does not usually compete with other 
propulsion devices. Examples of important applications are given below and 
discussed further in Chapter 4. 

Space Launch Vehicles 

Between the first space launch in 1957 and the end of 1998 approximately 4102 
space launch attempts have taken place in the world and all but about 129 were 
successful (see Ref. 1-20). Space launch vehicles or space boosters can be clas- 
sified broadly as expendable or recoverable/reusable. Other bases of classification 
are the type of propellant (storable or cryogenic liquid or solid propellants), num- 
ber of stages (single-stage, two-stage, etc.), size/mass of payloads or vehicles, 
and manned or unmanned. 

Each space launch has a specific space flight objective, such as an earth orbit 
or a moon landing. See Ref. 1-21. It uses between two and hve stages, each 
with its own propulsion systems, and each is usually fired sequentially after the 



1 .3. APPLICATIONS OF ROCKET PROPULSION 1 5 



lower stage is expended. The number of stages depends on the specific space 
trajectory, the number and types of maneuvers, the energy content of a unit mass 
of the propellant, payload size, and other factors. The initial stage, usually called 
the booster stage, is the largest and it is operated first; this stage is then separated 
from the ascending vehicle before the second-stage rocket propulsion system is 
ignited and operated. As will be explained in Chapter 4, adding an extra stage 
permits a significant increase in the payload (such as more scientific instruments 
or more communications gear). 

Each stage of a multistage launch vehicle is essentially a complete vehicle in 
itself and carries its own propellant, its own rocket propulsion system or systems, 
and its own control system. Once the propellant of a given stage is expended, the 
remaining mass of that stage (including empty tanks, cases, structure, instruments, 
etc.) is no longer useful in providing additional kinetic energy to the succeeding 
stages. By dropping off this useless mass it is possible to accelerate the final 
stage with its useful payload to a higher terminal velocity than would be attained 
if multiple staging were not used. Both solid propellant and liquid propellant 
rocket propulsion systems have been used for low earth orbits. 

Figure 1-12 shows the Delta IV heavy-lift space launch vehicle at takeoff. 
Its propellants are liquid oxygen/liquid hydrogen (LOX/LH 2 ) in all its main 
engines. Its booster engine, the Pratt & Whitney Rocketdyne RS-68, is shown 
in Fig. 6-10 and data is in Table 11-2 and its second-stage engine, the Pratt 
& Whitney Rocketdyne RL lOB-2 LOX/LH 2 (24,7501b thrust) is shown in Fig. 
8-17 and data is in Table 8-1. The two liquid propellant strap-on booster pods 
(with the same booster engine) are removed for smaller payloads. Figure 1-13 
shows the Atlas V space launch vehicle. Its booster engine is the Energomash 
RD-I80, it has Aerojet solid propellant strap-on boosters, and the upper stage 
engine is the Pratt & Whitney Rocketdyne RL lOA-4-2 LOX/LH 2 engine. The 
Russian (Energomash) LOX/kerosene RD-180 engine is shown in Ref. 1-2 as its 
Figure 7.10-1 1 and data is in its Table 7.10-2. In both of these two launch vehicles 
the payload is carried on top of the second stage and it has its own propulsion 
system of small thrusters. Table 1-3 gives data of the larger propulsion systems 
of these two U.S. launch vehicles. 

The U.S. Space Shuttle is one of the older programs and it provided the 
first reusable spacecraft that lands on a runway. Figure 1-14 shows the basic 
configuration of the Space Shuttle, which consists of two stages, the booster and 
the orbiter. It shows all the 67 rocket propulsion systems of the shuttle. The 
orbiter is really a reusable combination space launch vehicle, spacecraft, and 
glider for landing. The two solid propellant strap-on rocket motors are the largest 
in existence; they are equipped with parachutes for sea recovery of the bumed-out 
motors. The large LO 2 /LH 2 external tank is jettisoned and expended just before 
orbit insertion (see Ref. 1-22). Details of several of these Space Shuttle rocket 
propulsion systems are given in Table 1 -4. The Space Shuttle accomplishes both 
civilian and military missions of placing satellites in orbit, undertaking scientific 
exploration, supplying a space station, and repairing, servicing, and retrieving 
satellites. 



16 CLASSIFICATION 




FIGURE 1-12. Heavy lift Delta IV space launch vehicle. The center liquid propellant 
booster stage has a Pratt & Whitney Rocketdyne RS-68 rocket engine (LOX/LH 2 ). The 
two strap-on stages each use the same engine. (Courtesy Pratt & Whitney Rocketdyne) 




1.3. APPLICATIONS OF ROCKET PROPULSION 



17 




FIGURE 1-13. Atlas V space launch vehicle with three (or five) strap-on stages using 
Aerojet solid rocket motors and a central Energomash (Russia) RD-180 liquid propellant 
booster rocket engine running on LOX/kerosene. (Courtesy Lockheed Martin Corp.) 



TABLE 1-3. Propulsion System Data for Two Major U.S. Lauch Vehicles 



00 

C 

^ -2 s 

3 

(u 3 

S 



a 

S < 

o ti 

Z >< 

PJ 



D CD 

s s 

^ 2 

U (X 



h — 

.H 3 S 

D ^ p_, o 

3 3 o E 

X O ^ 



O D 

fj~ IZ3 , 

s a 

^ £ ' 
c/5 h3 



^ flj 

.2 S M 

tfl C cd 

3 -2 c/3 
S izi y 

Oh 



g -3 S 

“ I ^ 

s 2 

Q <=^ 






»/l 



o 

o 

CN 



»o 

cs 



o 



o 






VJ ■,' 

cd >-J 
> C/5 



o^ 

O 



C/5 

Cli 



CQ 

o 

-] 

0^: 



■a 

cd 

O 



X3 

2 



CN 

O 



00 



!C 00 

£ 

_ o 

m ^ 



CN 



'sd 



CN 

C CN 

Z m 



c/5 



o^ 

r- 



r- 

o 



T3 

C 

cd 



CO 



O 

O 

CQ 



Q 

Cii 



§ 

w 

Q 



on 

J 



r- 

»o 

r- 



o 



00 


< 


CN 




00 






1 


»4 


Z 


CN 


4.9 



o 

»o 









m 








CN 




cn 


m 








































Cj 






o 








Lh 




o 

cd 


Cj 


o 








cd 

> 


Cj 






o 




> 


'2 


cd 


00 


4= 










Z| 


o 






00 






o 




z 






is 


s 








CN 


Os 

in 


cd 

(D 




w 






'flj 




m 


0\ 


o 


00 






o 




c/3 

3 

SZ 


c 

'oo 

c 

w 




cn 


ci a 


::^ 




r- 

00 




in 


CN 

00 


OS 

OS 




o 

o 

o 


o S 

O 3 


o 

o 


o ^ 
8 5 






o 


cn 

o 


o 


H 


Lh 

(D 

a 


o 

D- 


in 


U-3 £ 

s 'a 


CN 

m 

'd- 


t"- 
m ' 


cn 

r-" 


(SL) 


o 

cn 


o 

CN 

o' 


o 

o 

m 






’d 




s 






00 




cn 


so 


CN 












CN 


CN 




Os 


00 


CN 



CN 

Lh 

o 



CN 

■4 

o 

Q 

c< 



18 



'(SL) means sea level. 

At ignition. 

■RD-180 has 2 gimbal mounted thrust chambers. 





19 



FIGURE 1-14. Simplified sketch of original version of Space Shuttle vehicle. The shuttle orbiter — the delta-winged vehicle about the size of 
a medium-range jet liner — is a reusable, cargo carrying, spacecraft-airplane combination that takes off vertically and lands horizontally like a 
glider. Each shuttle orbiter was designed for a minimum of 100 missions and can carry as much as 65,0001b of payload to a low Earth orbit, 
and a crew of up to four members and 10 passengers. It can return up to 25,0001b of payload back to earth. 




TABLE 1-4. Propulsion Systems for the Space Shuttle 



§ ^ 



.ii H 

^ c: 



z ^ 



o c = 
Z <u 

s- ° 



■Q cj 



£) 

C 

<rs 



6 -e 

w) o 

s 



-S 3 

00 T3 



U 1) 
3 — 

.S 3 

U ”3 



IS o 



C O 

■3 -a 



<D u (U £f' 



O- 



c- 

' x> 



-£ c o 



q:;' c u o 'co ^ 



r- 3 o =i 



2 ^ 



If 

-3 T3 

■g ■= 

'3 .S' 



00 

Z in 



'tT 



CO 3 in 

S •§ ji 



^ m 



o .«> 
O "O 



o -a o ■£ "S 



■3 s- ^ 

C _ o '3 O 

w S o -s 3 p 

>■ 52 is ^ ^ ^ 

— .3, = 'C ~o 

5 -2 R c 2 S 



o E 3 

r- S 

2£ > -7 

lis 

P o 

•P 



^ ^ ^ 
R 3 £ 



3 ^ 

°S ^ 



^ "S 

3 CO 
.3 (U 



2 >n 



o 3 : 

<U N 
cfl N 



m c 



^ CN 
O 



II R 



S 



I ■g § s o 



6 

o 

ifc 



!C 3 S -s ^ 

O £ .9- g 2 = 



— a a> 



>> H 



R 



« QS 
rT '*- 

■S £ 

3 

rs 

o 

O rn 

rn 



CM CN 

3 r4 



CQ 

Qi 



9" 

2 

a II 
-o ^ 



VO 

VO 



I ° 

£ 

£ - -a 
« 13 .S 



— 3 



XI .00 



“ -S 

ctS 2 
00 p 



20 



1. MMH, monomethylhydrazine and NTO, nitiogen tetroxide. 

2. 70% Ammonium perchlorate; 16% aluminum; 12% polybutadiene acrylic acid binder; 2% epoxy curing agent. 



1 .3. APPLICATIONS OF ROCKET PROPULSION 21 



Table 1-4 gives propulsion system data on the Space Shuttle, which is really 
a combination of launch vehicle, spacecraft, and a glider. It can be seen that 
the thrust levels are highest for booster or first stages and are relatively high for 
upper stages (thousands of pounds). Only for the attitude control system of the 
vehicle (also called reaction control in Table 1 -4) are the thrust levels low (from 
a fraction of a pound for small spacecraft to as high as about 1000 pounds thrust 
in the Space Shuttle vehicle). Frequent propulsion starts and stops are usually 
required in these applications. 

At the time of this writing (early 2008) for this new 8th edition National 
Aeronautics and Space Administration (NASA) had awarded the initial contracts 
for a new large manned space flight vehicle identified as Ares I. It is intended 
to replace the aging Space Shuttle after about 2012. It is planned to use a large 
single 5-segment solid rocket motor as booster propulsion (being developed by 
ATK Launch Systems) and the second stage will use a J-2X liquid propellant 
rocket engine with LOX/LH 2 propellants (being developed by Pratt & Whitney 
Rocketdyne). 

A new set of manned space flight vehicles are currently being developed by 
several entrepreneurial US and foreign companies, many with their own private 
capital. They are aimed at the future commercial market of sending tourists 
(for a hefty price) into space and returning them safely back to earth. All are 
based on reusable spacecraft, some with reusable launch vehicles, some with 
vertical and some with horizontal takeoff or landing. A couple of suhorbital flights 
have already been accomplished with professional pilots using a winged vehicle. 
Liquid propellant rocket engines seem to be preferred and many are planned to 
be reusable. It is too early for determining which of these organizations will be 
successful in commercializing manned space flight. 

The missions and payloads for space launch vehicles are many, such as 
military (reconnaissance satellites, command and control satellites), nonmili- 
tary government [weather observation satellites, global positioning system (GPS) 
satellites], space exploration (space environment, planetary missions), or com- 
mercial (communication satellites). 

A single stage-to-orbit vehicle, attractive because it avoids the costs and com- 
plexities of staging, has been expected to have improved reliability (simpler 
structures, fewer components). However, its payload is usually too small. A low 
earth orbit (say 100 miles altitude) can only be achieved with such a vehicle 
if the propellant performance is very high and the structure is efficient and low 
in mass. Liquid propellants such as liquid hydrogen with liquid oxygen were 
usually chosen. To date a large rocket-propelled single stage-to-orbit vehicle has 
not flown. 

Spacecraft 

Depending on their missions, spacecraft can be categorized as earth satellites, 
lunar, interplanetary, and trans-solar types, and as manned and unmanned 
spacecraft. Reference 1-23 lists over 20,000 satellites and categorizes them as 
satellites for communications, weather, navigation, scientific exploration, deep 



22 CLASSIFICATION 



space probes, observation (including radar surveillance), reconnaissance and 
other applications. Rocket propulsion is used for both primary propulsion (i.e., 
acceleration along the flight path, such as for orbit insertion or orbit change 
maneuvers) and secondary propulsion functions in these vehicles. Some of the 
secondary propulsion functions are attitude control, spin control, momentum 
wheel and gyro unloading, stage separation, and the settling of liquids in tanks. 
A spacecraft usually has a series of different rocket propulsion systems, some 
often very small. For spacecraft attitude control about three perpendicular axes, 
each in two rotational directions, the system must allow the application of pure 
torque for six modes of angular freedom, thus requiring a minimum of 12 
thrusters. Some missions require as few as 4 to 6 thrusters, whereas the more 
complex manned spacecraft have 40 to 80 thrusters in all of its stages. Often 
the small attitude control rockets must give pulses or short bursts of thrust, 
necessitating thousands of restarts. See Section 6.7 and Ref. 1-24. 

Table 1-5 presents a variety of spacecraft along with their weights, missions, 
and propulsion. Although only U.S. launch vehicles are listed in this table, there 
are also launch vehicles developed by France, the European Space Agency, Rus- 
sia, Japan, China, India, and Israel that have successfully launched payloads into 
satellite orbits. They use rocket propulsion systems that were developed in their 
own countries. 

The majority of spacecraft have used liquid propellant engines, a few with solid 
propellant boosters. Many spacecraft have operated successfully with electrical 
propulsion for attitude control. Electrical propulsion systems will probably also 
be used for some primary and secondary spacecraft propulsion missions on long- 
duration space flights, as described in Chapter 17. 

Micropropulsion is a relatively new designation. It has been deflned as any 
rocket propulsion system that is applicable to small spacecraft with a mass of 
less than 100 kg, or 220 lb. See Ref. 1-25. It encompasses a variety of different 
propulsion concepts, such as certain very low thrust liquid mono- and bipropellant 
rocket engines, small gaseous propellant rocket engines, several types of electrical 
propulsion systems, and emerging advanced versions of these. Many are based 
on fabrication of very small components (valves, thrusters, switches, insulators, 
or sensors) by micromachining and electromechanical processes. 

Missiles and Other Applications 

Military missiles can be classified as shown in Table 1-6. Rocket propulsion for 
new U.S. missiles uses now almost exclusively solid propellant rocket motors. 
They can be strategic missiles, such as long-range ballistic missiles (800 to 
9000 km range), which are aimed at military targets within an enemy country, or 
tactical missiles, which are intended to support or defend military ground forces, 
aircraft, or navy ships. 

Tables 1-6 and 1-7 show some parameters of rocket propulsion devices 
for different applications. The selection of the best rocket propulsion system 
type and design for any given application is a complex process involving many 
factors, including system performance, reliability, propulsion system size, and 
compatibility, as described in Chapter 19. 



Space Maneuver Propulsion 



t 

I 

ci 



-Q 



u 

3 



D 

C 

;> 



X) 

E 



D 

■a c 

C 

o 

in 

o 



V3 



T3 

C 



Oh •'S 
D X 



too 

•S 

’B 

c 



(U 2 

■y C 

CO 3 



■£ c 

cO C 
ja CO 



2 M 

u S 

^ C 

a . 



cO 



cO 

c 

D 



pH 

E 

D 



M c 

a o 

.2 

to 

CJ Dh 

X o 



o 

O X 

X D 

-e'o^ 

o 

T3 c/3 
CO ID 
0^: 0^ 



OJ) 

c ^ 

X o 

Q- 

^ (D 

D 2 

X .2 

g d . 

•c “ ^ 

CO d ^ 

• o 



c/3 

00 



c/i 



^ a 

U Q, 

,o 

'+^ c 

o 



’3 E 

Ci- o 



X 

.SP 

I 



0 


0 


0 


0 0 


0 


X- 


0 


0 


0 


0 


0 


0 


0 0 


0 


wo 


0 


00 


r- 




wo 


wo 


I> wo 


0 


00 


0 




0 






l> 




o' 


p 


wo" 




p 








X 


wo 




CN 







& 

o 

£ 



c c d c c 

cO cO cO cO cO 



D D D D 
Oh O- Oh D. 

O O O O 



O- O- 
O O 



Oh D- 
O O 



a 

2 

a 

o 

c c 

O D 
C (30 

e o 



o _ 



D D D D D X 

_C _C _C 

N N N N N . 

cO cO cO cO cO X 

l- ^ ^ 5- Vh D 

-O »!■ 
>> 2 
i 00 



T3 -a 

X E 



-a X 

X ffi 



X cs 

f 5 i ffi 
s i i s § 
^ ° Is s 

o § ^O o 

Z ^ 



<N (N 

' z z 



Oh 

O 

i-i 

Xh 

O 

C 

O 



K K E 



O 

cs 

z 



K 



a 

o 

Vh 

a 

o 

c 

o 

2 



-d X 

X E 



c 

o 

c 

D 

X 



CT 

X 



b s 



CO 

X 

c 

o 



•c 

D- 



3 

H 



CO 

Oh 

C/5 

C/5 

p 



D 

C/5 






D 

2 

CO 

Z 





^ ^ 


c/3 


S/J 


3 


0 

!>. 


C/i 






b-s 


L' 3 '3 


3 

3 


c/3 

*2 


X 


00 


'2 

3 


s_ 

3 

X 


-a 


3 c= 


g c C c 

§ c 0 0 3? 

2 ‘u 0 y _ 


X 


3 


0 


eg) 


'+-( 


3 


3 


s ® 

3 0 


CHh 


X 


0 

0 


c/3 


X 


0 

D 


0 

D 



o 

wo 



'C Dh <0 

o 8 p 

wo X wo 



D O 
^ O 
WO 

o 

O (N 



X 

c 

cO 

X 

c 

CO 

2 



OS 


o' 




2 


X 




WO 


0 


u 

D 


Uh 

D CJ) 


c/3 

3 


D 

0 


3 


2 c 


X 


0 


3 

S 


0 X 

E > 


2 

Z 


0 



o 

o 



o 



D 

I 

o 

D 



cO 

a 

C/5 



wo ^ 
(N ^ 



00 ^ 

CO c/5 O 



C 

3 

2 

2 

o 

U 



o ^ 
^ d 



c 

o 

D CJ 



CN 

O 



C/5 

Q 



D > 



PP 



O _ 
O D 

n , .H 



3 (i> 



3 

a 

c/5 

& 

D 



23 



‘'N2O4, nitrogen tetroxide (oxidizer); MMH, monomethylhydrazine (fuel); 50;50 UDMH-N2H4 is a 50% mixture of unsymmetrical dimethylhydrazine and 
hydrazine. 



TABLE 1-6. Selected Data for U.S. Missiles 



£ 


o 


o 


in 


00 


so 


00 o 


in 


o 


o 


CM 


o 




in 


o 




»n 


o 


, , 


o 


o 


00 


Os 


OS 


-H in 


r- 


o 


CO 


in 


00 


OS 


— H 


o 


CM 


os 


o 








1-^ 


CO 


Os^ 


00^ 


'd- 




CM^ 


1-H 


OS 




in 




'ct 




o 


_oo 


od' 

i> 


in 

so 






o 








cm" 












cm" 




o" 



C 

D 

cd 



c 


3 


3 


•o 


-o 


3 


TO 


■a 


T 3 


dJ 


C /2 

.s 


T 3 


dJ 


d 3 


d 5 


d 5 


3 

3 


d 3 


3 


o 


3 


3 




CO 


3 


id 


id 


id 


id 


cd 


id 


id 


id 


id 


id 


cr 


id 


3 


*02 


c /2 


c/2 


O 


o 


c/2 


o 


o 


o 


o 


bO 


o 


o 


o 


o 


o 


id 


o 


c/2 


3 


C/f 


C/f 


c/2 


c/2 


C/ 2 " 


c/2 


C/i 


c/2 


c/2 


■a 


c/2 


c/2 


C/i 


c/2 


c/2 


c/2 


C/i 


C/f 




a> 


(U 


o" 


o" 


(U 


o" 


o" 


o" 


o" 


o 


o" 


o" 


o" 


o" 


o" 


o 


o" 


(U 


o 


00 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


bO 


> 


s 


•2 


-2 


3 


s 


3 


iS 


3 


3 


3 


3 


3 


iS 


3 


3 


3 


3 


3 


Oh 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 


c/2 






(U 

s 

cd 

z 



o 

W) 

(D 

td 

u 

c 

o 



CO CM — — CN 



CM 



00 in »o 

OS ^ Os 'sd 

m ^ ^ 



CM 

CM CM ^ CM 
m:> SO d — ' 



r- 



O^^coOScMCMO 
— ^ -H ^ CM CM 



^ ^ O 
CM m o 



i> so in CM in 

sO'^socM^sor^co 



cd 

X 



c 

o 

T3 



S "a 

E 



.u- 

U .2 
TO in 
- W td 
c/5 Cu 



■d 



^ $ o 

• ^ < CJ 
J= Cd 

c/5 c/5 pin 



(D 

■| g o 

^ Oh D 
c/5 c/5 c/5 



D 

0J5 

C 

cd 

Ui 

00 

c 

o 



(U 
CJ 

^ ,cd 

Vh 4-( 

O id 

o I 3 

a ‘f 

■c S 2 

I 2 8 



HJ 

c 



»n CO 

d cd 



c 

cd 



00 
D c 

S-H .S 

CC ^ 



OJ 



D 

00 



D 

C/5 



24 



Tow (antitank) 0.58 3.84 1 stage, solid 40 

Cruise missile (subsonic) Tomahawk 1.74 21 solid booster + 3,900 

turbofan 



TABLE 1-7. Typical Propulsion Characteristics of Some Rocket Applications 



< 

a 

3 

B 



o 

<N 



3 00 

, O 

6 o 
■ 00 ^ 
o 2 
a. 



S' 



a. - 

D 



o 

ao lo 



c/5 D 



c 

•c 

£ 



1 

(N 



<N 

O 



a. 

D 



I 

o 



2 c/5 



B 

(N 



Q, 

ID 



C 

B 



I 

CM 

d 



3 

H 



O O 0) c/5 



-D 



^ -5 



^ .a 

flj ^ 

^ .s 

D. c/!l 
c/5 flJ 



<u 



Cu 6D 



O 

U 



2 

s: 



c 

'5b 



T3 

a 



o 

•c 



2 

C ■“ 

'c^ 






D 

'3 00 



.ti 1) 

5 .2 



^ C/5 

'5 ^ 

O' o 

J 



e — ^ 

3 3 — 



O O 
C/5 C/5 



7D 

3 '3 

3 O' "O 

2 3 

00 c/5 



O o 

^ 2 
C gj 
3 

1j o 

0“ S 

, o i2 



Q. U c/5 

o a -3 
c o 5 

o >- S' 

S s 



'O 

3 2 
o s 

.3 c/5 

k. 

. 3 O 

p (S 



s 

<N 

o 

3 

'5 

O' 



3 Q. C 

3 



3 O' 

^ ;3 



U 



o 

u. 

Br 

3 



3 

3 



'O = = 



o- 2 c 



o. 

< 



0) o 
00 g 
fa Xi 



§■■2 
d G 
c/5 < 



3 .3 3 Q 



02 



3 ^ 

os 



c 

< 



25 



'go is acceleration of gravity at the Earth’s surface = 9.8066 m/sec^ or 32.17 ft/sec^. 



26 CLASSIFICATION 



The term surface launch can mean a launch from the ground, the ocean surface 
(from a ship), or from underneath the sea (submarine launch). Some tactical 
missiles, such as the air-to-surface short-range attack missile (SRAM), have a 
two-pulse solid propellant motor, where two separate, insulated grains are in the 
same motor case; the time interval before starting the second pulse can be timed to 
control the flight path or speed profile. Most countries now have tactical missiles 
in their military inventories, and many of these countries have a capability to 
produce their own rocket propulsion systems that are used to propel them. 

Solid propellant rocket motors are being used today in most tactical missiles, 
such as in surface-to-air, air-to-air, air-to-surface, and surface-to-surface appli- 
cations, for the ejection of pilot aircraft seats or crew capsules, target drones, 
signal rockets, weather sounding rockets, antitank rockets, or for the separations 
of stages in a multistage flight vehicle. 

Applications, which were popular 30 to 60 years ago, but are no longer active, 
include liquid propellant rocket engines for propelling military fighter aircraft, 
assisted takeoff rocket engines and rocket motors, and superperformance rocket 
engines for augmenting the thrust of an aircraft jet engine. 



REFERENCES 

1-1. E. C. Goddard and G. E. Pendray (Eds), The Papers of Robert H. Goddard, 
three volumes, McGraw-Hill Book Company, 1970, 1707 pages. It includes 
the pioneering treatise “A Method of Reaching Extreme Altitudes” originally 
published as Smithsonian Miscellaneous Collections, Vol. 71, No. 2, 1919. 

1-2. G. P. Sutton, History of Liquid Propellant Rocket Ensines, published by AIAA, 
2006, 911 pages. 

1-3. B. N. Yur’yev (Ed), Collected Works of K. E. Tsiolkowski, Vols. 1-3, USSR 
Academy of Sciences, 1951; also NASA Technical Translation F-236, April 1965. 

1 -4. H. Oberth, Die Rakete zu den Planet enrdumen (By Rocket to Planetary Space), 
R. Oldenburg, Munich, 1923 (in German), a classical text. 

1-5. E. Sanger, Raketenflugtechnik (Rocket Flight Technology), R. Oldenburg, Munich, 
1923 (in German). 

1-6. W. von Braun and F. Ordway, History of Rocketry and Space Travel, 3rd ed., 
Thomas Y. Crowell, New York, 1974. 

1-7. L. H. Caveny, R. L. Geisler, R. A. Ellis, and T. L. Moore, “Solid Enabling 
Technologies and Milestones in the USA,” Journal of Propulsion and Power, Vol. 
19, No. 6, Nov.-Dee. 2003, AIAA, pp. 1038-1066. 

1-8. A. M. Lipanov, “Historical Survey of Solid Propellant Rocket Development in 
Russia,” Journal of Propulsion and Power, Vol. 19, No. 6. Nov. -Dec. 2003, pp. 
1063-1088. 

1-9. AGARD Lecture Series 150, Design Methods in Solid Rocket Motors, 
AGARD/NATO, Paris, April 1988. 

1-10. A. Davenas, Solid Rocket Propulsion Technology , Pergamon Press, London (orig- 
inally published in French), 1988. 



REFERENCES 27 



1-11. S. V. Gunn and C. M. Ehresman, “The Space Propulsion Technology Base 
Established Four Decades Ago for the Thermal Nuclear Rocket Is Ready for 
Current Application,” AIAA paper 2003-4590, July 2003. 

1-12. R. W. Bussard and R. D. DeLauer, Nuclear Rocket Propulsion, McGraw-Hill 
Book Company, New York, 1958. 

1-13. D. Buden, “Nuclear Rocket Safety,” Acta Astronautica, Vol. 18, 30 Years of 
Progress in Space, 1988, pp. 217-224. 

1-14. R. C. Finke (Ed), Electric Propulsion and Its Application to Space Missions, 
Vol. 79, Progress in Aeronautics and Astronautics Series, AIAA 1981. 

1-15. R. G. Jahn, Physics of Electrical Propulsion, McGraw-Hill Book Company, New 
York, 1968, 339 pages. 

1-16. T. Svitek et al., “Solar Sails as Orbit Transfer Vehicle — Solar Sail Concept Study,” 
Phase II Report, AIAA paper 83-1347, 1983. 

1-17. V. P. Ageev et al., “Some Characteristics of the Laser Multi-pulse Explosive Type 
Jet Acta Astronautica, Vol. 8, No. 5-6, 1981, pp. 625-641. 

1-18. R. A. Hyde, “A Laser Fusion Rocket for Interplanetary Propulsion,” Preprint 
UCRL 88857, Lawrence Livermore National Laboratory, Livermore, California, 
Sept. 1983. 

1-19. NASA International Space Station (A resource on the ISS by NASA; includes 
operational use, wide range of background material, archives, image gallery and 
planned missions) www.nase.gov/station-34k. 

1-20. T. D. Thompson (Ed), TRW Space Log , Vols. 3 1 to 34, TRW Space and Electronics 
Group (today part of Northrop Grumman Corp.), Redondo Beach, CA, 1996 and 
1997-1998. 

1-21. S. J. Isakowitz, J. B. Hopkins, and J. P. Hopkins, International Reference Guide 
to Space Launch Systems, 4th ed., AIAA, 2004, 596 pages. 

1 -22. National Aeronautics and Space Administration (NASA), National Space Trans- 
portation System Reference, Vol. 1, Systems and Eacilities, U.S. Government 
Printing Office, Washington, DC, June 1988; a description of the Space Shuttle. 

1-23. A. R. Curtis (Ed), Space Satellite Handbook, 3rd ed, Gulf Publishing Company, 
Houston, TX, 1994, 346 pages. 

1-24. G. P. Sutton, History of Small Liquid Propellant Thrusters, presented at the 52nd 
JANNAF Propulsion Meeting, May 2004, Las Vegas, NE, published by the Chem- 
ical Propulsion Information Analysis Center, Columbia, Maryland, June 2004. 

1-25. M. M. Micci and A. D. Ketsdever, Micropropulsion for Small Spacecraft, Progress 
in Aeronautics and Astronautics Series, Vol. 187, AIAA, 2000, 477 pages. 



CHAPTER 2 



DEFINITIONS AND FUNDAMENTALS 



The basic principles of rocket propulsion are essentially those of mechanics, 
thermodynamics, and chemistry. Propulsion is achieved by applying a force to 
a vehicle, that is, accelerating the vehicle or, alternatively, maintaining a given 
velocity against a resisting force. This propulsive force is obtained by ejecting 
propellant at high velocity. This chapter deals with the definitions and the basic 
relations of this propulsive force, the exhaust velocity, and the efficiencies of 
creating and converting the energy and other basic parameters. The symbols 
used in the equations are defined at the end of the chapter. Wherever possible 
the American Standard letter symbols for rocket propulsion (as given in Ref. 
2-1) are used. 



2.1. DEFINITIONS 

The total impulse It is the thrust force F (which can vary with time) integrated 
over the burning time t: 



For constant thrust and very short start and stop transients this reduces to 



Impulse If is proportional to the total energy released by all the propellant in a 
propulsion system. 




( 2 - 1 ) 



It = Ft 



( 2 - 2 ) 



28 



2.1. DEFINITIONS 29 



The specific impulse is the total impulse per unit weight of propellant. It is 
an important figure of merit of the performance of a rocket propulsion system, 
similar in concept to the miles per gallon parameter used with automobiles. 
A higher number means better performance. Values of Is are given in many 
chapters of this book, and the concept of an optimum specific impulse for a 
particular mission is introduced later. If the total mass flow rate of propellant is 
m and the standard acceleration of gravity at sea level go is 9.8066 m/sec^ or 
32.174 ft/sec^, then 



IpFdt 

gojrhdt 



(2-3) 



This equation will give a time-averaged specific impulse value for any rocket 
propulsion system, particularly where the thrust varies with time. During tran- 
sient conditions (during start or the thrust buildup period, the shutdown period, 
or during a change of flow or thrust levels) values of /* can be obtained by inte- 
gration or by determining average values for F and m for short time intervals. 
For constant thrust and propellant flow this equation can be simplified; below, 
Mp is the total effective propellant mass expelled through the nozzle. 



h = It/impgo) (2-4) 

In Chapter 3 there is further discussion of the specific impulse. For constant 
propellant mass flow m, constant thrust F, and negligibly short start or stop 
transients: 



Is = F/(mgo) = F/w 

hKmpgo) = It/w (2-5) 

at or near the earth’s surface, the product ffipgo is the total effective propellant 
weight w, and the weight flow rate is w. But in space or outer satellite orbits, 
the mass is just multiplied by an arbitrary constant, namely go- In the Systeme 
International (SI) or metric system of units 4 can be expressed simply in “sec- 
onds” because of the use of the constant go- In the United States today we still 
use the English Engineering (EE) system of units (foot, pound, second) in many 
of the chemical propulsion engineering, manufacturing, and test operations. In 
many past and current U.S. publications, data, and contracts, the specific impulse 
has units of thrust (Ibf) divided by weight flow rate of propellants (Ibf/sec), 
simplified as seconds. The numerical value of 4 is the same in the EE and 
the SI system of units. However, the units of Is do not represent a measure of 
elapsed time but a thrust force per unit “weight” flow rate. In this book the 
symbol Is is used for the specific impulse, as listed in Ref. 2-1. Eor solid pro- 
pellant and several other rocked systems the symbol fip is often used, as listed in 
Ref. 2-2. 



30 DEFINITIONS AND FUNDAMENTALS 



In a rocket nozzle the actual exhaust velocity is not uniform over the entire 
exit cross section and does not represent the entire thrust magnitude. The velocity 
profile is difficult to measure accurately. For convenience a uniform axial velocity 
c is assumed, which allows a one-dimensional description of the problem. This 
ejfective exhaust velocity c is the average equivalent velocity at which propellant 
is ejected from the vehicle. It is defined as 

C = hgQ = F /ih (2-6) 

It is given either in meters per second or feet per second. Since c and /j differ 
only by an arbitrary constant, either one can be used as a measure of rocket 
performance. In the Russian literature c is generally used in lieu of /* . 

In solid propellant rockets it is difficult to measure the propellant flow rate 
accurately. Therefore, the specific impulse is often calculated from total impulse 
and the propellant weight (using the difference between initial and final motor 
weights and Eq. 2-5). In turn the total impulse is obtained from the integral of the 
measured thrust with time, using Eq. 2-1. In liquid propellant units it is possible 
to measure thrust and instantaneous propellant flow rate and thus to use Eq. 2-3 
for calculation of specific impulse. Equation 2-4 allows another definition for 
specific impulse, namely, fhe amount of impulse imparted to a vehicle per unit 
sea-level weight of propellant expended. 

The term specific propellant consumption refers to the reciprocal of the specific 
impulse and is not commonly used in rocket propulsion. It is used in automotive 
and duct propulsion systems. Typical values are listed in Table 1-2. 

The mass ratio IVR of a vehicle or a particular vehicle stage is defined to 
be the final mass mj (after rocket operation has consumed all usable propel- 
lant) divided by mo (before rocket operation). The various terms are depicted in 
Fig. 4-1. 

IVR = m//mo (2-7) 

This applies to a single or a multistage vehicle; for the latter, the overall mass 
ratio is the product of the individual vehicle stage mass ratios. The final mass 
m/ is the mass of the vehicle after the rocket has ceased to operate when all 
the useful propellant mass nip has been consumed and ejected. The final vehicle 
mass mf includes all those components that are not useful propellant and may 
include guidance devices, navigation gear, payload (e.g., scientific instruments 
or a military warhead), flight control systems, communication devices, power 
supplies, tank structure, residual or unusable propellant, and all the propulsion 
hardware. In some vehicles it can also include wings, fins, a crew, life support 
systems, reentry shields, landing gears, and the like. Typical values of IVR can 
range from 60% for some tactical missiles to less than 10% for some unmanned 
launch vehicle stages. This mass ratio is an important parameter in analyzing 
flight performance, as explained in Chapter 4. When IVR is applied to a single 
stage, then its upper stages become the “payload.” 



2.1. DEFINITIONS 31 



The propellant mass fraction ^ indicates the fraction of propellant mass 
in an initial mass mq. It can he applied to a vehicle, a stage of a vehicle, or to a 
rocket propulsion system. 



^ = Mp/mo ( 2 - 8 ) 

C = (mo - mf)lmQ = mp/{mp + mj) (2-9) 

mo = mf + lUp ( 2 - 10 ) 

When applied to a rocket propulsion system, the mass ratio IVR and propellant 
fraction (; are different from those that apply to a vehicle as described above. 
Here the initial or loaded mass mo consists of the inert propulsion mass (the 
hardware necessary to burn and store the propellant) and the effective propellant 
mass. It would exclude masses of nonpropulsive components, such as payload or 
guidance devices. For example, in a liquid propellant rocket engine the final or 
inert propulsion mass m/ would include the propellant feed tanks, their pressur- 
ization system (with turbopump and/or gas pressure system), one or more thrust 
chambers, various piping, fittings and valves, an engine mount or engine struc- 
ture, filters, and some sensors. The residual or unusable remaining propellant 
is usually considered to be part of the final inert mass mf, as it will be in this 
book. However, some rocket propulsion manufacturers and some literature assign 
residuals to be part of the propellant mass mp . When applied to a rocket propul- 
sion system, the value of the propellant mass fraction indicates the quality of 
the design; a value of, say, 0.91 means that only 9% of the mass is inert rocket 
hardware, and this small fraction contains, feeds, and burns a substantially larger 
mass of propellant. A high value of f is desirable. 

The impulse-to-weight ratio of a complete propulsion system is defined as the 
total impulse f divided by the initial or propellant-loaded vehicle weight wq. A 
high value indicates an efficient design. Under our assumptions of constant thrust 
and negligible start and stop transients, it can be expressed as 



Wo (m/ -F mp)go 
= 

mf/mp + 1 



( 2 - 11 ) 

( 2 - 12 ) 



The thrust-to-weight ratio F /wq expresses the acceleration (in multiples of 
the Earth’s surface acceleration of gravity) that the engine is capable of giving 
to its own loaded propulsion system mass. For constant thrust the maximum 
value of the thrust-to-weight ratio, or maximum acceleration, occurs just before 
termination or burnout because the vehicle mass has been diminished by the mass 
of useful propellant. Values of F /w are given in Table 2-1. The thrust-to-weight 
ratio is useful to compare different types of rocket systems. 



TABLE 2-1. Ranges of Typical Performance Parameters for Various Rocket Propulsion Systems 



3 1 

H 



3 

'B 

.3 BO 

D. C 

I 



“ I 

Q. O 
c^CIh 



00 



a _ 

.2 § 

CA 

Oh U 



cd 

.s 

00 



D 

t3 

t-H 

OhO 

e 



u 



^ 3 D 

^ CIh 



Gh 

<D 

C 

*ob 

c 

PJ 



00 



pp 



x: 

00 



PP 



c c 

(D (U 
> > 
o o 



c 

D 

S 

s D « -H 
g- Oh Oh Oh 

5 G- 

o o x: 4= 

> 5P op 



x: 

00 



c 

o 

g : 

ID 

> 

x: 



o c 

:9 3 5 

5 "o 

!3r tA Gh ^ 



pH PP 



^ E 

Z |_3’ <N 

. X E 

<N (N 

X z 



x: 

00 



pp 



S 

Gh 

"d 

> 

D 

T3 



Dh 

Dh 



D D <N 



E X Z E 



I 

o 







































r «~) 




1 






X 






O 










o 




o 






















— H 




"-H 


— H 


^H 


in 


"-H 


^H 


1 






r4 




1 




1 


1 


1 


1 


1 


1 








1 

r4 








m 


m 


m 


- 


m 


cs 


O 






o 




o 




1 

O 


1 

O 


1 

O 


o 


1 

O 


O 








d 




















c3 


























O 

CA 

"O 




(A 

D 


Q 

Vi 

T3 


CA 

D 


2 

A 

T3 


CA 

D 
















3 


3 


3 






Weeks 


Weeks 


CA 




c 

o 

D 

D 

00 


few 


3 

'§ 


3 

O 

D 

D 

C/5 


3 

s 


3 

O 

D 

D 

CO 


3 

E 


Days 


Days 


4 Ch 

3 

O 

S 


Days 








cs 










CS 








CS 


o 






1 








1 


1 


1 




1 


1 


o 






O 




o 




o 


O 


o 




O 


O 


1 






1 




m 

1 




1 


1 


1 




1 


1 


1 

cs 






1 




1 

(N 




1 

(N 


1 


1 

so 


Tl - 


1 

so 


1 

m 


o 






O 




o 




o 


o 


o 


o 


O 


1 

O 




























o 


























o 






o 


























o 










o 
















CO 

1 




o 




o 


o 








o 


1 








o 




o 


o 


1 


1 


1 


o 


1 






1 




o 




OS 




1 


1 


1 


m 


o 






o 




(N 




04 


o ' 










o 






o 






(N 
















so 




















04 
















o 


o 


o 


o 




00 






m 




o 




O 


o 


o 


o 


o 


o 








(N 




so 




o 


CN 


in 


r - 


o 


o 


1 






r4 

1 




00 

1 




1 


1 


r4 

1 


1 


in 

1 


I > 

1 


1 

o 






1 




1 

o 




1 

o 


1 

o 


1 

o 


1 

o 


1 

o 


1 

o 


o 






o \ 




o 




in 


00 


o 


o 


o 


o 


04 






HH 




in 




1-H 


(N 


I > 


o 


r4 








































































13 










IH 
















3 


00 








'3 
















D 


.2 








cr 






4—1 




















x : 






3 

E 










O 




SI 






u 

o 






"d 










Vh 

D 


13 

.2 


G 


D 

B 

CA 

o 




IH 

’o 

CA 


c 

cO 




Gh 

2 

Gh 

O 




3 

.2 






E 

"d 

1 


D 

D 

c 


3 

2 

CA 

3 


00 




1 j 




3 

o 




CA 

A 

S3 




^ , 


00 

.3 


00 

o3 


’ Gh 

O 


f -, 

D 


.2 


13 

D 

’2 


G - 

o 

Vh 

G 




s 

E 




E 




D 

-g 

CA 


3 

D 

4 = 


2 

o 

u 


-3 ,D 
D 

E ^ 


D 

'd 

1 


D 

4 = 

u 


D 

4 = 

u 


z 




3 

O " 




Ij 

3 

Z 




CA 

D 

OC 


D 


D 

D 

s 


3 _i 

- E3 

X 


Ion - 


E 

’o 

cn 



G 3 



CS D 
C ^ 

.2 e 

CA D 



Q. S' 

D C 



3 ^ 

s ^ 

o ' ^ 

7D O 



'cA t2 

^ II 

o ^ 
^ 2 
o ^ 



0.-S 

C3 

< Qi 



32 



Kinetic power per unit exhaust mass flow. 



2.2. THRUST 33 



Example 2-1. A rocket projectile has the following characteristics: 



Initial mass 

Mass after rocket operation 
Payload, nonpropulsive structure, etc. 
Rocket operation duration 
Average specific impulse of propellant 



200 kg 
130 kg 
110 kg 
3.0 sec 
240 sec 



Determine the mass ratios for the overall vehicle and for the propulsive unit, the 
propellant mass fraction for the propulsive unit, the effective exhaust velocity, and the total 
impulse. Also, sensitive electronic equipment in the payload would limit the maximum 
acceleration to 35go’s. Is this exceeded during the flight? Assume constant thrust and 
neglect the short start and stop transients. 

SOLUTION. From the givens, the mass of the expended propellant is 70 kg. The mass 
ratio (Eq. 2-8) for the vehicle is therefore IVR = mf/mo = 130/200 = 0.65, and the the 
propulsion unit is (130 — 110)/(200 — 110) = 0.222. Note that these are different. Now 
the propellant mass fraction of the propulsion system is 



Note that this mass fraction is acceptable for tactical missiles but might be only “fair” 
for spacecraft. Because the effective exhaust velocity is proportional to the specific 
impulse, c = 240 x 9.81 = 2354.4 m/sec. The propellant mass flow rate is m = 70/3 = 
23.3 kg/sec and the thrust becomes (Eq. 2-5) 



Now the total impulse can be calculated as /, = 54936 x 3 = 164,808 N-sec. The 
impulse-to- weight ratio (Eq. 2-11) for the propulsion system at the earth’s surface is 
I,/wq = 164,808/[(200 — 110)9.81] = 187 sec (this reflects only a fair design because 
Is = 240 sec). 

For a horizontal trajectory, the maximum acceleration is found at the end of the 
thrusting schedule just before shutdown (because while the thrust is unchanged the mass 
is now at its minimum value of 130 kg). 



This represents 43.08 go’s, at the earth’s surface and exceeds the limit of the 
equipment. 



2.2. THRUST 

The thrust is the force produced by a rocket propulsion system acting upon a 
vehicle. In a simplified way, it is the reaction experienced by its structure due to 
the ejection of matter at high velocity. It represents the same phenomenon that 



C = 70/(200- 110) = 0.778 



F = 23.3 X 2354.4 = 54,936 N 



Final acceleration = F fm = 54,936/130 = 422.58 m/sec^ 



34 DEFINITIONS AND FUNDAMENTALS 



pushes a garden hose backwards or makes a gun recoil. In the latter case, the 
forward momentum of the bullet and the powder charge is equal to the recoil 
or rearward momentum of the gun barrel. Momentum is a vector quantity and 
is defined as the product of mass times velocity. All ship propellers and oars 
generate their forward push at the expense of the momentum of the water or air 
masses, which are accelerated toward the rear. Rocket propulsion differs from 
these devices primarily in the relative magnitude of the accelerated masses and 
velocities. In rocket propulsion relatively small masses are involved that are 
carried within the vehicle and ejected at high velocities. 

The thrust, due to a change in momentum, is given below. A derivation can be 
found in earlier editions of this book. The thrust and the mass flow are constant 
and the gas exit velocity is uniform and axial. 

dm w 

F = —V2 = mv2 = —V2 (2-13) 

dt go 

This force represents the total propulsion force when the nozzle exit pressure 
equals the ambient pressure. 

The pressure of the surrounding fluid (i.e., the local atmosphere) gives rise to 
the second contribution that influences the thrust. Figure 2-1 shows schematically 
the external pressure acting uniformly on the outer surface of a rocket chamber 
and the changing gas pressures on the inside of a typical thermal rocket engine. 
The size and length of the arrows indicates the relative magnitude of the pressure 



Atmosphere Converging nozzle section 




FIGURE 2-1. Pressure balance on chamber and nozzle interior walls is not uniform. The 
internal gas pressure (indicated by length of arrows) is highest in the chamber (pi) and 
decreases steadily in the nozzle until it reaches the nozzle exit pressure p 2 - The external 
or atmospheric pressure p 3 is uniform. At the throat the pressure is /?,. The four subscripts 
(shown inside circles) refer to the quantities A, v, T, and p at specific locations. 



2.2. THRUST 35 



forces. The axial thrust can be determined by integrating all the pressures acting 
on areas that can be projected on a plane normal to the nozzle axis. The forces 
acting radially outward are appreciable but do not contribute to the axial thrust 
because a rocket is typically an axially symmetric chamber. The conditions prior 
to entering the nozzle are essentially stagnation conditions. 

Because of a fixed nozzle geometry and changes in ambient pressure due to 
variations in altitude, there can be an imbalance of the external environment 
or atmospheric pressure and the local pressure p 2 of the hot gas jet at the 
exit plane of the nozzle. Thus, for a steadily operating rocket propulsion system 
moving through a homogeneous atmosphere, the total thrust is equal to (this 
equation is arrived at using the control volume approach in gas dynamics; see 
Refs. 2-3 and 2-4); 



F = mv2 + (P2 - Pi)A2 (2-14) 

The first term is the momentum thrust represented by the product of the pro- 
pellant mass flow rate and its exhaust velocity relative to the vehicle. The second 
term represents the pressure thrust consisting of the product of the cross-sectional 
area at the nozzle exit A 2 (where the exhaust jet leaves the vehicle) and the differ- 
ence between the exhaust gas pressure at the exit and the ambient fluid pressure. 
If the exhaust pressure is less than the surrounding fluid pressure, the pressure 
thrust is negative. Because this condition gives a low thrust and is undesirable, 
the rocket nozzle is usually so designed that the exhaust pressure is equal or 
slightly higher than the ambient fluid pressure. 

When the ambient atmosphere pressure is equal to the exhaust pressure, the 
pressure term is zero and the thrust is the same as in Eq. 2-13. In the vacuum 
of space P 3 =0 and the thrust becomes 



F = mv2+P2Ai (2-15) 

At the altitude where the exhaust pressure is exactly equal to the surrounding 
fluid pressure (p 2 = P$) the pressure thrust is zero. For any nozzle this occurs 
only at one attitude, and this is referred to as the rocket nozzle with optimum 
nozzle expansion ratio A 2 /A;. This is further elaborated in Chapter 3. 

Equation 2-14 shows that the thrust of a rocket unit is independent of the 
flight velocity. Because changes in ambient pressure affect the pressure thrust, 
there is a variation of the rocket thrust with altitude. Because atmospheric pressure 
decreases with increasing altitude, the thrust and the specific impulse will increase 
as the vehicle is propelled to higher altitudes. This change in pressure thrust due 
to altitude changes can amount to between 10 and 30% of the overall thrust, as is 
shown for a typical rocket engine in Fig. 2-2. Table 8-1 shows the sea level and 
high-altitude thrust for several rocket engines. Appendix 2 gives the properties 
of the standard atmosphere (ambient pressure p^). 



36 DEFINITIONS AND FUNDAMENTALS 




FIGURE 2-2. Altitude performance of RS 27 liquid propellant rocket engine used in 
early versions of the Delta launch vehicle. 



2.3. EXHAUST VELOCITY 

The ejfective exhaust velocity as defined by Eq. 2-6 applies to all rockets that 
thermodynamically expand hot gas in a nozzle and, indeed, to all mass expul- 
sion systems. From Eq. 2-14 and for constant propellant mass flow this can be 
modified to 



c = V2 + {p2- Pz)^ilm ( 2 - 16 ) 

Equation 2-6 shows that c can be determined from thrust and propellant flow 
measurements. When p 2 = Pi, the effective exhaust velocity c is equal to the 
average actual exhaust velocity of the propellant gases V 2 - When p 2 ^ Pi, then 
c zfz V 2 - The second term of the right-hand side of Eq. 2-16 is usually small 
in relation to V 2 , thus the effective exhaust velocity is usually close in value to 
the actual exhaust velocity. When c = V 2 , the thrust (from Eq. 2-14) can be 
rewritten as 



F = (w I go)v 2 = the (2-17) 

The characteristic velocity has been used frequently in the rocket propulsion 
literature. Its symbol c*, pronounced “cee-star,” is defined as 

c*=piAt!m (2-18) 

The characteristic velocity c* is used in comparing the relative performance of 
different chemical rocket propulsion system designs and propellants; it is easily 
determined from measured data of m, p\, and At. It relates to the efficiency of 
the combustion and is essentially independent of nozzle characteristics. 



2.3. EXHAUST VELOCITY 37 



However, the specific impulse 4 and the effective exhaust velocity c are 
functions of the nozzle geometry, such as the nozzle area ratio A 2 /A,, as shown 
in Chapter 3. Some values of 4 and c* are given in Tables 5-5 and 5-6. 

Example 2-2. The following measurements were made a sea-level test of a solid pro- 
pellant rocket motor (all cross sections are circular and unchanging): 



Bum duration 


40 sec 


Initial propulsion system mass 


1210 kg 


Mass of rocket motor after test 


215 kg 


Sea-level thrust 


62,250 N 


Chamber pressure 


7.00 MPa 


Nozzle exit pressure 


70.0 kPa 


Nozzle throat diameter 


8.55 cm 


Nozzle exit diameter 


27.03 cm 



Determine m, V 2 , c* , and c at sea level. Also, determine the pressure thrust and specific 
impulse at sea level, 1000 m, and 25,000 m altitude. Assume the momentum thrust is 
invariant during the ascent, and the start and stop transients are short. 

SOLUTION. The cross sections corresponding to the given diameters are A, = 0.00574 
and A 2 = 0.0574 m^ (i.e., an area ratio of 10). The mass flow rate for all altitudes is 

m = (1210 - 215)/40 = 24.88 kg/sec 

The desired c* (all altitudes) and c at sea level follow (Eqs. 2-6 and 2-18): 

c* = piA,lm = 7.00 X 106 x 0.00574/24.88 = 1615 m/sec 
c = F/in = 62250/24.88 = 2502 m/sec 

The pressure thrust at sea level is 

(pi -P3)A2 = (0.070 - 0.1013) X 10® X 0.0574 = -1797 N 
so the nozzle exit velocity becomes (all altitudes), from Eq. 2-14, 

V 2 = (62250 - 1797)/24.88 = 2430 m/sec 

The remaining answers are shown in the table below. We obtain the ambient pressures 
from Appendix 2. 



Altitude 


P 3 (kPa) 


Pressure Thrust (N) 


4 (sec) 


Sea level 


101.32 


-1798 


248 


1000 m 


89.88 


-1141 


250 


25,000 m 


2.55 


3871 


111 



Further examination of the standard atmosphere table in Appendix 2 reveals that the 
pressure thrust becomes zero at an elevation of just under 3000 m. 



38 DEFINITIONS AND FUNDAMENTALS 



2.4. ENERGY AND EFFICIENCIES 

Although efficiencies are not commonly used directly in designing rocket units, 
they permit an understanding of the energy balance of a rocket system. Their 
definitions are arbitrary, depending on the losses considered, and any consistent 
set of efficiencies, such as the one presented in this section, is satisfactory in 
evaluating energy losses. As stated previously, two types of energy conversion 
processes occur in any propulsion system, namely, the generation of energy, 
which is really the conversion of stored energy into available energy and, subse- 
quently, the conversion to the form in which a reaction thrust can be obtained. 
The kinetic energy of ejected matter is the form of energy useful for propulsion. 
The power of the jet Pjet is the time rate of expenditure of this energy, and for a 
constant gas ejection velocity v this is a function of f and P: 



The term specific power is sometimes used as a measure of the utilization 
of the mass of the propulsion system including its power source; it is the jet 
power divided by the loaded propulsion system mass, Pjet /mo. For electrical 
propulsion systems that carry a heavy, relatively inefficient energy source, the 
specific power can be much lower than that of chemical rockets. The energy input 
from the energy source to the rocket propulsion system has different forms in 
different rocket types. For chemical rockets the energy is created by combustion. 
The maximum energy available per unit mass of chemical propellants is the heat 
of the combustion reaction Qr ; the power input to a chemical engine is 



where 7 is a conversion constant that depends on the units used. A large portion of 
the energy of the exhaust gases is unavailable for conversion into kinetic energy 
and leaves the nozzle as residual enthalpy. This is analogous to the energy lost 
in the hot exhaust gases of internal combustion engines. 

The combustion efficiency for chemical rockets is the ratio of the actual and 
the ideal heat of reaction per unit of propellant and is a measure of the source 
efficiency for creating energy. Its value is high (approximately 94 to 99%), and it 
is defined in Chapter 3. When the power input Pchem is multiplied by the combus- 
tion efficiency, it becomes the power available to the propulsive device, where it 
is converted into the kinetic power of the exhaust jet. In electric propulsion the 
analogous efficiency is the power conversion efficiency. For solar cells it has a 
low value; it is the efficiency for converting solar radiation energy into electric 
power (10 to 20%). 

The power transmitted to the vehicle at any one time is defined in terms of 
the thrust of the propulsion system F and the vehicle velocity u : 



Fjet = jthv^ = jWgol^ — jFgof = jFV2 



(2-19) 



^chem — 



( 2 - 20 ) 



^vehicle — 



(2-21) 



2.4. ENERGY AND EFFICIENCIES 39 



The internal efficiency of a rocket propulsion system is an indication of the effec- 
tiveness of converting the system’s energy input to the propulsion device into 
the kinetic energy of the ejected matter; for example, for a chemical unit it is the 
ratio of the kinetic power of the ejected gases expressed by Eq. 2-19 divided by 
the power input of the chemical reaction as given in Eq. 2-20. Internal efficien- 
cies are used in Example 2-4. The energy balance diagram for a chemical rocket 
(Eig. 2-3) shows typical losses. The internal efficiency can be expressed as 



'lint = 



kinetic power in jet 
available chemical power 



'Icomb^chem 



( 2 - 22 ) 



Typical values of jjint are listed later in Example 2-4. 

An object moving through a fluid affects the fluid (i.e., stirs it) in ways that 
may hinder its motion and/or require extra energy expenditures. This is one 
consequence of skin friction and can be substantial. The propulsive efficiency 
(Eig. 2-4) reflects this energy cost for rocket vehicles. The equation that deter- 
mines how much of the kinetic energy of the exhaust jet is useful for propelling 
a rocket vehicle is defined as 

vehicle power 

rjp = 

vehicle power -f residual kinetic jet power 
Fu 2u fc 

= = (2-23) 

Em - h im(c - m)^ 1 -f- ( m/c)^ 

where F is the thrust, u the absolute vehicle velocity, c the effective rocket 
exhaust velocity with respect to the vehicle, m the propellant mass flow rate, and 
rjp the propulsive efficiency. The propulsive efficiency is a maximum when the 



Heat loss to walls 



Combustion loss 
(poor mixing, 
incomplete burning) 







f 


100% 




Unavailable thermal 
energy of exhaust jet 



Residual kinetic energy 
of exhaust gases 
^0to50% 



40 to 70% 



LUseful energy for 
vehicle propulsion 
— Kinetic energy of exhaust jet 
-Total energy of exhaust jet 
-Available energy in combustion chamber 
-Heating value of propellants 



FIGURE 2-3. Typical energy balance diagram for a chemical rocket. 



40 DEFINITIONS AND FUNDAMENTALS 




0 1.0 2.0 3.0 

Velocity ratio, u/c 

FIGURE 2-4. Propulsive efficiency at varying velocities. 

forward vehicle velocity is exactly equal to the exhaust velocity. Then the residual 
kinetic energy and the absolute velocity of the jet are zero and the exhaust gases 
stand still in space. 

While it is desirable to use energy economically and thus have high efficien- 
cies, there is also the problem of minimizing the expenditure of ejected mass, 
which in many cases is more important than minimizing the energy. In nuclear 
reactor energy and some solar energy sources, for example, there is an almost 
unlimited amount of heat energy available; yet the vehicle can only carry a limited 
amount of working fluid. Economy of mass expenditures of working fluid can be 
obtained if the exhaust velocity is high. Because the specific impulse is propor- 
tional to the exhaust velocity, it is a measure of this propellant mass economy. 



2.5. MULTIPLE PROPULSION SYSTEMS 

The relationships below are for determining the overall (oa) or total thrust and the 
overall mass flow of propellants for a group of propulsion systems (two or more) 
bring in parallel (i.e., in the same direction at the same time). These relationships 
apply to liquid propellant rocket engines, solid propellant rocket motors, electrical 
propulsion systems, hybrid propulsion systems, and to any combinations of these. 
Many space launch vehicles and larger missiles have multiple propulsion systems. 
The Space Shuttle, for example, has three larger liquid engines and two large 
solid motors bring jointly at liftoff. 

The overall thrust, Eoa, is needed for determining the bight path, and the overall 
mass bow rate, moa, is needed for determining the vehicle’s mass decrease with 
time. Together these two determine the overall specibc impulse, iIs)on' 

= = Fi+F2 + F3 + ... (2-24) 

h?oa = = nil + m2 + m + . . . (2-25) 

(4)oa = ^ F/(go ^ m) 



( 2 - 26 ) 



2.6. TYPICAL PERFORMANCE VALUES 41 



For liquid propellant rocket engines with a turbopump and a gas generator, 
there is a separate turbine outlet flow that is usually dumped overboard through 
a pipe and a nozzle (see Fig. 1 -4), which needs to be included in the equations 
above. 

Example 2-3. The MA-3 multiple liquid engine for the Atlas missile has two booster 
engines (F = 165,000 Ibf at sea level and m = 66.71bm/sec each). The turbine exhaust 
gas of both boosters provides an extra 2300 Ibf of thrust at a propellant flow rate of 
about 3.3 Ibm/sec. These booster engines are dropped from the vehicle after about 145 
sec operation. The central sustainer rocket engine, which also starts at liftoff, continues 
to operate for 300 sec. It has a sea-level thrust of 57{,j000 Ibf at a mass flow of 27.05 
Ibm/sec. Its turbine exhaust gases are aspirated into the nozzle of the engine. There are 
also two small vernier engines used for roll control of the vehicle during the sustainer-only 
portion of the flight. They each have a thrust of 415 Ibf at sea level with a propellant flow 
of 0.213 Ibm/sec. Determine the overall thrust Foa and the overall mass flow rate moa at 
liftoff, when all the nozzles are pointing vertically down. 

SOLUTION. Use Eqs. 2-24 and 2-25. The thrust from the turbine must be included in 
2-24. 

Foa = 165,000 X 2 -h 2300 x 2 -h 57,000 -h 415 x 2 = 390,430 Ibf 

moa = 66.7 X 2 -h 3.3 X 2 -h 27.05 -h 0.213 x 2 = 167.5 Ibm/sec 



2.6. TYPICAL PERFORMANCE VALUES 

Typical values of representative performance parameters for different types of 
rocket propulsion are given in Table 2-1 and in Fig. 2-5. 

Chemical rockets have relatively low values of specific impulse, relatively light 
machinery (i.e., low engine weight), a very high thrust capability, and therefore 
high acceleration and high specific power. At the other extreme, the ion propul- 
sion devices have a very high specific impulse, but they must carry a heavy 
electrical power source with them to deliver the power necessary for high ejec- 
tion velocities. The very low acceleration potential for the electrical propulsion 
units and those using solar radiation energy usually requires a long period for 
accelerating and thus these systems are best used for missions where the flight 
time is long. The low thrust values of electrical systems imply that they are not 
useful in fields of strong gravitational gradients (for takeoff or landing) but are 
best used in a true space flight mission. 

The chemical systems (solid and liquid propellant rockets) are fully developed 
and widely used for many different vehicle applications. They are described in 
Chapters 5 to 16. Electrical propulsion has been in operation in many space 
flight applications (see Chapter 17). Some of the other types are still in their 
exploratory or development phase, but may become useful. 

Example 2-4. Compare the types of propulsion system tabulated below by computing 
the power input and propellant flow required for each to deliver 106.8 N (25 Ibf) of thrust. 



42 DEFINITIONS AND FUNDAMENTALS 




10 ® 10 ® 10 '' 10 ® 10 2 10 1 1 10 100 1000 
Acceleration in multiples of earth gravity Qq or thrust to 
vehicle \«eight ratio 



FIGURE 2-5. Exhaust velocities as a function of typical vehicle accelerations. Regions 
indicate approximate performance values for different types of propulsion systems. The 
mass of the vehicle includes the propulsion system, but the payload is assumed to be 
zero. 



SOLUTION. From Eqs. 2-13 and 2-19 and assuming that V 2 = c. 



m = F/I,go 



Power input = Pjet/hint = V2 mvllmm 



Engine Type 


hint 


Is (sec) 


V 2 (m/sec) 


in (kg/sec) 


Pinput(kW) 


Chemical rocket 


0.70 


300 


2,943 


[0.0363] 


[225] 


Nuclear fission 


0.50 


800 


7,848 


[0.0136] 


[838] 


Arc-electrothermal 


0.40 


600 


5,886 


[0.0181] 


[785] 


Ion electrostatic 


0.65 


2000 


19,620 


[0.0054] 


[1612] 



Results are shown inside square brackets in the table. More than half a megawatt input is 
needed for the last three propulsion systems but note that the propellant flows are small. 
These data are only illustrative because to date there are no single arc or ion thrusters 
that supply the required thrust so multiple units would be required. Electric propulsion is 
presently restricted to thrust levels of less than one newton in part because of the lack 



PROBLEMS 43 



of such large electrical power supplies. Nuclear propulsion units are not available at the 
present time so chemical propulsion would remain as the only option. Note that it would 
be more realistic to compare total impulse because thrust times can vary between types. 



Variable Thrust 

There are flight missions that require a change in thrust during flight. Some solid 
rocket propellant motors can have a fixed predetermined thrust usually consisting 
of an initial short duration high-thrust boost phase followed by a lower thrust 
longer sustaining phase. Details may be found in Section 12.3. Some liquid 
propellant rocket engines can have random variable thrust that can be adjusted 
during flight. Details are in Section 8.7. In small low-thrust liquid propellant 
attitude control thrusters, it is possible to control the number and/or timing of 
repeated short pulses to achieve a variable thrust effect. This concept is discussed 
in Section 8.3. 

A predetermined thrust decrease (used in some solid propellant rocket motors) 
can be obtained with a special design of the grain allowing a high-thrust period 
(for missile acceleration) followed by a low-thrust period for maintaining the 
flight velocity against atmospheric drag. Alternatively, it may be a special two- 
pulse motor with two different grain compositions in the same combustion cam- 
ber, one for a high-thrust boost phase and one for a subsequent low-thrust 
sustaining phase. Both of these are discussed in Chapter 12. 

Some solid propellant and liquid propellant experimental propulsion systems 
have used variable nozzle throat areas (achieved with a variable position “tapered 
pintle” at the nozzle throat) and one experimental version has flown. To date, there 
has been no published information on the further production and implementation 
of such systems. 



PROBLEMS 

When solving problems, three appendixes (see end of book) may be helpful: 

Appendix 1. Conversion Factors and Constants 
Appendix 2. Properties of the Earth’s Standard Atmosphere 
Appendix 3. Summary of Key Equations 

1 . A jet of fluid hits a stationary flat plate in the manner shown below. 




44 DEFINITIONS AND FUNDAMENTALS 



(a) If there is 50 kg of fluid flowing per minute at an absolute velocity of 200 m/sec, 
what will be the force on the plate? 

Answer: 167 N. 

(b) What will this force be when the plate moves in the direction of flow at n = 50 
km/h? Explain the methodology. 

Answer: 144 N. 

2. The following data are given for a certain rocket unit: thrust, 8896 N; propellant 
consumption, 3.867 kg/sec; velocity of vehicle, 400 m/sec; energy content of propellant, 
6.911 Ml/kg. Assume 100% combustion efficiency. 

Determine (a) the effective velocity; (b) the kinetic jet energy rate per unit flow of 
propellant; (c) the internal efficiency; (d) the propulsive efficiency; (e) the overall 
efficiency; (f) the specific impulse; (g) the specific propellant consumption. 

Answers: (a) 2300 m/sec; (b) 2.645 MJ/kg; (c) 38.3%; (d) 33.7%; (e) 13.3%; 
(f) 234.7 sec; (g) 0.00426 sec”'. 

3. A certain rocket has an effective exhaust velocity of 7000 ft/sec; it consumes 280 
Ibm/sec of propellant mass, each of which liberates 2400 Btu/lbm. The unit operates 
for 65 sec. Construct a set of curves plotting the propulsive, internal, and overall 
efficiencies versus the velocity ratio ujciQ < ujc < 1.0). The rated flight velocity 
equals 5000 ft/sec. Calculate (a) the specific impulse; (b) the total impulse; (c) the 
mass of propellants required; (d) the volume that the propellants occupy if their average 
specific gravity is 0.925. 

Answers: (a) 217.5 sec; (b) 3,960,000 Ibf-sec; (c) 18,200 Ibm; (d) 315 ftl 

4. For the rocket in Problem 2, calculate the specific power, assuming a propulsion system 
dry mass of 80 kg and a duration of 3 min. 

5. A Russian rocket engine (RD-110) consists of four nonmoveable thrust chambers 
supplied by a single turbopump. The exhaust from the turbine of the turbopump then 
drives four vernier chamber nozzles (which can be rotated to provide some control of 
the flight path). Using the information below, determine the thrust, effective exhaust 
velocity, and mass flow rate of the four vernier thrusters. 

Individual thrust chambers (vacuum): 

F = 73.14 kN,c = 3279 m/sec 
Overall engine with verniers (vacuum): 

F = 297.93 kN,c = 3197 m/sec 

Answers: 5.358 kN, 1351 m/sec, 3.965 kg/sec. 

6. A certain rocket engine has a specific impulse of 250 sec. What range of vehicle 
velocities {u, in units of ft/sec) would keep the propulsive efficiencies at or greater 
than 80%. Also, how could rocket-vehicle staging be used to maintain these high 
propulsive efficiencies for the range of vehicle velocities encountered during launch? 
Answers: 4021 to 16,085 ft/sec; design upper stages with increasing /j. 



SYMBOLS 45 



7. Plot the variation of the thrust and specific impulse against altitude, using the atmo- 
spheric pressure information given in Appendix 2, and the data for the Minuteman 
first-stage rocket thrust chamber in Table 12-3. Assume that p 2 = 8.66 psia. 

8 . During the boost phase, the three Space Shuttle main engines (SSMEs) operate together 
with the two solid propellant rocket motors (SRBs) for the first 2 minutes. For the 
remaining thrust time, the SSMEs operate alone. Using Table 1-4, calculate the overall 
specific impulse for the vehicle during the 2-minute combined thrust operation. 
Answer: 310 sec. 

9. For the values given in Table 2-1 for the various propulsion systems, calculate the 
total impulse for a fixed propellant mass of 2000 kg. 



SYMBOLS (English engineering units are given in parentheses) 

A area, m^(ft^) 

At nozzle throat area, m^(ft^) 

A 2 exit area of nozzle, m^(ft^) 

c effective velocity, m/sec (ft/sec) 

c* characteristic velocity, m/sec (ft/sec) 

E energy, J (ft-lbf) 

F thrust force, N (Ibf) 

For overall force, N (Ibf) 

go standard sea-level acceleration of gravity, 

9.80665 m/sec2(32.174 ft/sec^) 

Is specific impulse, sec 

Us) OR overall specific impulse, sec 

It impulse or total impulse, N-sec (Ibf-sec) 

J conversion factor or mechanical equivalent of heat, 4.184 J/cal or 

1055 J/Btu or 778 ft-lbf/Btu 

m mass, kg (slugs) (1 slug = mass of 32.174 lb of weight at sea 

level) 

nioR overall mass, kg 

m mass flow rate, kg/sec (Ibm/sec) 

nif final mass (after rocket propellant is ejected), kg (Ibm or slugs) 

nip propellant mass, kg (Ibm or slugs) 

mo initial mass (before rocket propellant is ejected), kg (Ibm or slugs) 

IVR mass ratio (m/-/mo) 

p pressure, pascal [Pa] or N/m^(lbf/ft^) 

P 3 ambient or atmospheric pressure, Pa (Ibf/ft^) 

P 2 rocket gas pressure at nozzle exit, Pa (Ibf/ft^) 

Pi chamber pressure, Pa (Ibf/ft^) 

P power, J/sec (ft-lbf/sec) 

Pj specific power, J/sec-kg (ft-lbf/sec-lbm) 



46 DEFINITIONS AND FUNDAMENTALS 



Qr heat of reaction per unit propellant, J/kg (Btu/lbm) 

1 time, sec 

u vehicle velocity, m/sec (ft/sec) 

V2 gas velocity leaving the rocket, m/sec (ft/sec) 

w weight, N or kg-m/sec^ (Ibf) 

w weight flow rate, N/sec (Ibf/sec) 

Wo initial weight, N or kg-m/sec^ (Ibf) 

Greek Letters 

^ propellant mass fraction 

r] efficiency 

>?comb combustion efficiency 

>7int internal efficiency 

r]p propulsive efficiency 



REFERENCES 

2-1. “American National Standard Letter Symbols for Rocket Propulsion”, ASME Pub- 
lication Y 10.14, 1959. 

2-2. “Solid Propulsion Nomenclature Guide,” CPIA Publication 80 , Chemical Propulsion 
Information Agency, Johns Hopkins University, Laurel, MD, May 1965, 18 pages. 
2-3. P. G. Hill and C. R. Peterson, Mechanics and Thermodynamics of Propulsion, 
Addison- Wesley, Reading, MA, 1992. 

2-4. R. D. Zucker and O. Bihlarz, Fundamentals of Gas Dynamics, 2nd ed., John Wiley 
& Sons, Hoboken, NJ, 2002. 



CHAPTER 3 



NOZZLE THEORY AND 
THERMODYNAMIC RELATIONS 



Thermodynamic relations of the processes inside a rocket nozzle and chamber 
furnish the mathematical tools needed to calculate the performance and deter- 
mine several of the key design parameters of rocket propulsion systems. They 
are useful as a means of evaluating and comparing the performance of various 
rocket systems; they permit the prediction of the operating performance of any 
rocket unit that uses the thermodynamic expansion of a gas and the determination 
of several necessary design parameters, such as nozzle size and generic shape, 
for any given performance requirement. This theory applies to chemical rocket 
propulsion systems (both liquid and solid and hybrid propellant types), nuclear 
rockets, solar-heated and resistance or arc-heated electrical rocket systems, and 
to any propulsion system that uses the expansion of a gas as the propulsive 
mechanism for ejecting matter at high velocity. 

These thermodynamic relations, which are fundamental and important in anal- 
ysis and design of rocket units, are introduced and explained in this chapter. The 
utilization of these equations should give the reader a basic understanding of 
the thermodynamic processes involved in rocket gas behavior and expansion. A 
knowledge of elementary thermodynamics and fluid mechanics on the part of the 
reader is assumed (see Refs. 3-1, 3-2, and 3-3). This chapter also addresses 
different nozzle configurations, nonoptimum performance, energy losses, noz- 
zle alignment, variable thrust, and four different ways for establishing nozzle 
performance parameters. 



47 



48 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



3.1. IDEAL ROCKET 

The concept of ideal rocket propulsion systems is useful because the relevant 
basic thermodynamic principles can be expressed as simple mathematical rela- 
tionships, which are given in subsequent sections of this chapter. These equations 
theoretically describe a quasi-one-dimensional nozzle flow, which corresponds to 
an idealization and simplification of the full two- or three-dimensional equations 
and the real aerothermochemical behavior. However, with the assumptions and 
simplifications stated below, they are very adequate for obtaining useful solutions 
to many rocket propulsion systems and preliminary design tasks. For chemical 
rocket propulsion the measured actual performance is usually between 1 and 
6% below the calculated ideal value. In designing new rockets, it has become 
accepted practice to use ideal rocket parameters that can then be modified by 
appropriate corrections, such as those discussed in Section 3.5. An ideal rocket 
unit is one for which the following assumptions are valid: 

1. The working substance (or chemical reaction products) is homogeneous . 

2. All the species of the working fluid are gaseous. Any condensed phases 
(liquid or solid) add a negligible amount to the total mass. 

3. The working substance obeys the perfect gas law. 

4. There is no heat transfer across the rocket walls; therefore, the flow is 
adiabatic. 

5. There is no appreciable friction and all boundary layer effects are 
neglected. 

6. There are no shock waves or discontinuities in the nozzle flow. 

7. The propellant flow is steady and constant. The expansion of the working 
fluid is uniform and steady, without vibration. Transient effects (i.e., start- 
up and shutdown) are of very short duration and may be neglected. 

8. All exhaust gases leaving the rocket have an axially directed velocity. 

9. The gas velocity, pressure, temperature, and density are all uniform across 
any section normal to the nozzle axis. 

10. Chemical equilibrium is established within the rocket chamber and the gas 
composition does not change in the nozzle (frozen flow). 

11. Stored propellants are at room temperature. Cryogenic propellants are at 
their boiling points. 

These assumptions permit the derivation of a simple, quasi-one-dimensional 
theory as mentioned in subsequent sections. Later in this book we present more 
sophisticated theories or introduce correction factors for several of the items on 
the list, and they allow a more accurate determination of the simplified analysis. 
The next paragraph explains why these assumptions cause only small errors. 

For a liquid bipropellant rocket the idealized theory postulates an injection 
system in which the fuel and oxidizer are mixed perfectly so that a homogeneous 
working substance results. A good rocket injector can approach this condition 



3.2. SUMMARY OF THERMODYNAMIC RELATIONS 49 



closely. For a solid propellant rocket unit, the propellant must essentially be 
homogeneous and uniform and the burning rate must be steady. For nuclear, 
solar-heated or arc-heated rockets, it is assumed that the hot gases are uniform 
in temperature at any cross section and steady in flow. Because chamber tem- 
peratures are typically high (2500 to 3600 K for common propellants), all gases 
are well above their respective saturation conditions and actually follow the per- 
fect gas law very closely. Assumptions 4, 5, and 6 above allow the use of the 
isentropic expansion relations in the rocket nozzle, thereby describing the max- 
imum conversion of heat to kinetic energy of the jet. This also implies that the 
nozzle flow is thermodynamically reversible. Wall friction losses are difficult to 
determine accurately, but they are usually negligible in nozzles. Except for very 
small chambers, the energy lost as heat to the walls of the rocket is usually less 
than 1% (occasionally up to 2%) of the total energy and can therefore often be 
neglected. Short-term fluctuations of the steady propellant flow rate and pressure 
are usually less than 5% of the rated value; their effect on rocket performance 
is small and can be neglected. In well-designed supersonic nozzles, the conver- 
sion of thermal energy into directed kinetic energy of the exhaust gases proceeds 
smoothly and without normal shocks or discontinuities; thus the flow expansion 
losses are generally small. 

Some companies and some authors do not include all or the same 11 items 
listed above in their definition of an ideal rocket. For example, instead of assump- 
tion 8 (all nozzle exit velocity is axially directed), some use a conical exit nozzle 
with a 15° half-angle as their base configuration in their ideal nozzle; this dis- 
counts the divergence losses, which are described later in this chapter. 



3.2. SUMMARY OF THERMODYNAMIC RELATIONS 

In this section we review briefly some of the basic relationships needed for the 
development of the nozzle flow equations. Rigorous derivations and discussions 
of these relations can be found in many thermodynamics or fluid dynamics texts, 
such as Refs. 3-1 and 3-2. 

The principle of conservation of energy can be readily applied to the adiabatic, 
no shaft-work process inside the nozzle. Furthermore, without shocks or friction, 
the flow entropy change is zero. The concept of enthalpy is useful in flow systems; 
the enthalpy comprises the internal thermal energy plus the flow work (or work 
performed by the gas at a velocity v in crossing a boundary). For ideal gases 
the enthalpy can conveniently be expressed as the product of the specific heat Cp 
times the absolute temperature T (the specific heat at constant pressure is formally 
defined as the partial derivative of the enthalpy with respect to temperature at 
constant pressure). Under the above assumptions, the total or stagnation enthalpy 
per unit mass ho is constant, that is, 



hf) = h + jU = constant 



(3-1) 



50 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



In the above, J is the mechanical equivalent of heat that is inserted only when 
thermal units (i.e., the Btu and calorie) are mixed with mechanical units (i.e., 
the ft-lbf and the joule). In SI units (kg, m, sec) the value of J is one. In the 
EE system of units the value of the constant J is given in Appendix 1. The 
conservation of energy for isentropic flow between any two sections x and y 
shows that the decrease in enthalpy or thermal content of the flow appears as an 
increase of kinetic energy since any changes in potential energy may be neglected. 

K - hy = i (u2 - vj)/J = Cp (T, -Ty) (3-2) 

The principle of conservatism of mass in a steady flow with a single inlet and 
single outlet is expressed by equating the mass flow rate m at any section x to 
that at any other section y ; this is known in mathematical form as the continuity 
equation. Written in terms of the cross-sectional area A, the velocity v, and the 
specific volume V, 



rhx = thy = m = AvjV (3-3) 

The perfect gas law is written as 

PxVx=RTx (3-4) 

where the gas constant R is found from the universal gas constant R' divided 
by the molecular mass 071 of the flowing gas mixture. The molecular volume 
at standard conditions becomes 22.41 m^/kg-mol or ft^/lb-mol and it relates to 
a value of R' = 8314.3 J/kg-mol-K or 1544 ft-lbf/lb-mol-°R. One often finds 
Eq. 3-3 written in terms of density p which is the reciprocal of the specific 
volume V. The specific heat at constant pressure Cp, the specific heat at constant 
volume c„, and their ratio k are constant for perfect gases over a wide range of 
temperatures and are related: 



k = Cp/Cy (3 -5a) 

Cp — Cy = R/J (3-5b) 

Cp = kR/ik - l)J (3-6) 

Eor an isentropic flow process the following relations hold between any points x 
and y: 



Tx/Ty = = iVy/Vx)^-^ (3-7) 

During an isentropic nozzle expansion the pressure drops substantially, the 
absolute temperature drops somewhat less, and the specific volume increases. 
When a flow is stopped isentropically the prevailing conditions are known as 
stagnation conditions and are designated by the subscript 0. Sometimes the word 



3.2. SUMMARY OF THERMODYNAMIC RELATIONS 51 



“total” is used instead of stagnation. As can be seen from Eq. 3-1 the stagnation 
enthalpy consists of the sum of the static or local enthalpy and the fluid kinetic 
energy. The stagnation temperature Tq is found from the energy equation as 

To = r + v'^UlCpJ) (3-8) 

where T is the absolute fluid static temperature. In adiabatic flows, the stagnation 
temperature remains constant. The relationship of the stagnation pressure to the 
local pressure in the flow can be found from the previous two equations: 

po/p = [1 + = {V/Vof (3-9) 

When the local velocity comes close to zero, the local temperature and pressure 
will approach the stagnation pressure and stagnation temperature. In a combus- 
tion chamber, where the gas velocity is small, the local combustion pressure 
is essentially equal to the stagnation pressure. The velocity of sound a or the 
acoustic velocity in ideal gases is independent of pressure. It is defined as 

a = sfm (3-10) 

In the EE system the value of R has to be corrected and the constant go is 
added. Equation 3-10 becomes y/g^kRT. This correction factor must be applied 
wherever R is used in EE units. The Mach number M is a dimensionless flow 
parameter and is used to define the ratio of the local flow velocity v to the local 
acoustic velocity a: 



M = v/a = v/sf^ (3-11) 

A Mach number less than one corresponds to subsonic flow and greater than 
one to supersonic flow. When the Mach number is equal to one, then the flow 
is moving at precisely the velocity of sound. It is shown later that at the throat 
of all supersonic nozzles the Mach number must be equal to one. The relation 
between stagnation temperature and Mach number can now be written from Eqs. 
3-2, 3-7, and 3-10 as 



TQ = T[\ + \{k-l)M^] 



(3-12) 



or 



M = 





Tq and pq designate the stagnation values of the temperature and pressure. Unlike 
the temperature, the stagnation pressure during an adiabatic nozzle expansion 
remains constant only for isentropic flows. It can be computed from 

P(,=p[\ + \{k-m^t'^'~'^ 



(3-13) 



52 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



The expansion area ratio for a nozzle with isentropic flow can be expressed 
in terms of Mach numbers for any points x and y within the nozzle. This rela- 
tionship, along with those for the ratios T /Tq and p/po, is plotted in Fig. 3-1 
for Ax = At and Mx = 1.0. Otherwise, 



A;, 



My^ 



1 -F [(^ - 1)/2]M,2 ] 
1 + [(^ _ 1)/2]M,2 I 



(3-14) 



As can be seen from Fig. 3-1, for subsonic flow the chamber contraction ratio 
AxjAf can be small, with values of 3 to 6, and the passage is convergent. There is 
no noticeable effect from variations of ^ . In solid rocket motors the chamber area 
A I refers to the flow passage or port cavity in the virgin grain. With supersonic 
flow the nozzle section diverges and the area ratio becomes large very quickly; 




' 0.10 1.0 10 



5 



o 

to 



Mach number 



FIGURE 3-1. Relationship of area ratio, pressure ratio, and temperature ratio as func- 
tions of Mach number in a De Laval nozzle for the subsonic and supersonic nozzle 
regions. 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 53 



the area ratio is significantly influenced by the value of k. The area ratio A 2 /A, 
ranges between 15 and 30 at M = 4, depending on the value of On the other 
hand, pressure ratios depend little on k whereas temperature ratios show more 
variation. 

The average molecular mass 371 of a mixture of gases is the sum of all the 
molar fractions n, multiplied by the molecular mass of each chemical species 
(n,371,) and then divided by the sum of all molar mass fractions. This is further 
discussed in Chapter 5. The symbol 371 is used to avoid confusion with M for the 
Mach number. In many pieces of rocket literature 371 is called molecular weight. 

Example 3-1. An ideal rocket is designed to operate at sea level using a propellant 
whose products of combustion have a specific heat ratio k of 1.3. Determine the required 
chamber pressure if the exit Mach number is 2.52. Also determine the nozzle area ratio 
between the throat and exit. 

SOLUTION. For optimum expansion the nozzle exit pressure must equal the local atmo- 
spheric pressure, which here has a value of 0.1013 MPa. If the chamber velocity may be 
neglected, then the ideal chamber pressure is the total stagnation pressure, which can be 
found from Eq. 3-13 as 



P0=p[1 + 3(^-1)M2]'''^<^ 

= 0.1013[1 -h 0.15 X 2.52^]‘-^l°-^ = 1.84 MPa 

The ideal nozzle area is determined from Eq. 3-14 setting M, = 1.0 at the throat (see 
also Eig. 3-1): 



M 

At 



1 

W2 



l + [{k- l)/2]Ml 
(k + l)/2 



(i-M)/2(*:-l) 



1 

2J2 



1 -hO.15 X 2.52^ 



2.3/0.6 



1.15 



3.02 



Note that ideal implies no losses, whereas optimum is a separate concept reflecting the 
best calculated performance at a particular set of given pressures. Optimum performance 
is often taken as the design condition and it occurs when p 2 = P 3 as will be shown in the 
section on the thrust coefficient (the peak of the curves in Eigs. 3-7 and 3-8 for fixed 
Pi/Pi)- 



3.3. ISENTROPIC FLOW THROUGH NOZZLES 

In a converging-diverging nozzle a large fraction of the thermal energy of 
the gases in the chamber is converted into kinetic energy. As will be explained, 
the gas pressure and temperature drop dramatically and the gas velocity can reach 
values in excess of 2 miles per second. This is a reversible, essentially isentropic 
flow process and its analysis is described here. If a nozzle inner wall has a flow 
obstruction or a wall protrusion (a piece of weld splatter or slag), then the kinetic 



54 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



gas enery is locally converted back into thermal energy essentially equal to the 
stagnation temperature and stagnation pressure in the chamber. Since this would 
lead quickly to a local overheating and failure of the wall, nozzle inner walls 
have to be smooth without any protrusion. Stagnation conditions can also occur 
at the leading edge of a jet vane (described in Chapter 18) or at the tip of a gas 
sampling tube inserted into the flow. 

Velocity 

From Eq. 3-2 the nozzle exit velocity i >2 can be found: 



This equation applies to ideal and nonideal rockets. For constant k this expres- 
sion can be rewritten with the aid of Eqs. 3-6 and 3-7. The subscripts 1 and 2 
apply to the nozzle inlet and exit conditions, respectively: 



This equation also holds for any two points within the nozzle. When the chamber 
section is large compared to the nozzle throat section, the chamber velocity or 
nozzle approach velocity is comparatively small and the term can be neglected. 
The chamber temperature Tj is at the nozzle inlet and, under isentropic conditions, 
differs little from the stagnation temperature or (for a chemical rocket) from the 
combustion temperature. This leads to an important simplified expression of the 
exhaust velocity V 2 , which is often used in the analysis: 



It can be seen that the exhaust velocity of a nozzle is a function of the pressure 
ratio pi/p 2 , the ratio of specific heats k, and the absolute temperature at the nozzle 
inlet Ti, as well as the gas constant R. Because the gas constant for any particular 
gas is inversely proportional to the molecular mass Tl, the exhaust velocity or 
the specific impulse are a function of the ratio of the absolute nozzle entrance 
temperature divided by the molecular mass, as is shown in Fig. 3-2. This ratio 
plays an important role in optimizing the mixture ratio in chemical rockets. 




(3- 15a) 




(3 -15b) 





(3-16) 



3.3. ISENTROPIC FLOW THROUGH NOZZLES 55 







I I I I 1_ 

50 75 100 125 150 

Tl/On, K-kg-mol/kg 

FIGURE 3-2. Specific impulse and exhaust velocity of an ideal rocket at optimum 
nozzle expansion as functions of the absolute chamber temperature T\ and the molecular 
mass Tl for several values of k and pi/p 2 - 



Equations 2-14 and 2-15 give the relations between the velocity V 2 , the 
thrust F, and the specific impulse Is', it is plotted in Fig. 3-2 for two pressure 
ratios and three values of k. Equation 3-16 indicates that any increase in the gas 
temperature (usually caused by an increase in energy release) or any decrease of 
the molecular mass of the propellant (usually achieved by using light molecular 
mass gases rich in hydrogen content) will improve the performance of the rocket; 
that is, they will increase the specific impulse Is or the exhaust velocity V 2 or 
c and, thus, the performance of the vehicle. The influences of the pressure ratio 
across the nozzle p\/p 2 and of the specific heat ratio k are less pronounced. As 
can be seen from Fig. 3-2, performance increases with an increase of the pressure 
ratio; this ratio increases when the value of the chamber pressure p\ increases 
or when the exit pressure p 2 decreases, corresponding to high altitude designs. 
The small influence of k values is fortuitous because low molecular masses are 
found in diatomic or monatomic gases, which have the higher values of k. 

For comparing specific impulse values from one rocket system to another or 
for evaluating the influence of various design parameters, the value of the pressure 
ratio must be standardized. A chamber pressure of 1000 psia (6.894 Mpa) and 
an exit pressure of 1 atm (0.1013 Mpa) are generally in use today. 



56 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



For optimum expansion p 2 — P 3 and the effective exhaust velocity c 
(Eq. 2-16) and the ideal rocket exhaust velocity are related, namely 



f2 = (c)opt (3-17) 

and c can be substituted for V 2 in Eqs. 3-15 and 3-16. For a fixed nozzle exit 
area ratio, and constant chamber pressure, this optimum condition occurs only 
at a particular altitude where the ambient pressure p^ happens to be equal to the 
nozzle exhaust pressure p 2 - At all other altitudes c ^ V 2 - 

The maximum theoretical value of the nozzle outlet velocity is reached with 
an infinite expansion (exhausting into a vacuum): 

(V2)max = V^kRTolik - 1) (3-18) 

This maximum theoretical exhaust velocity is finite, even though the pressure 
ratio is infinite, because it represents the finite thermal energy content of the fluid. 
Such an expansion does not happen, because, among other things, the temperature 
of many of the working medium species will fall below their liquefaction or the 
freezing points; thus they cease to be a gas and no longer contribute to the gas 
expansion. 

Example 3-2. A rocket operates near sea level with a chamber pressure of p\ = 2.068 
MPa or 300 psia, a chamber temperature of 2222 K, and a propellant consumption of 
m = 1.0 kg/sec. Take k = 1.30 and R = 345.7 J/kg-K. Calculate the ideal thmst and the 
ideal specific impulse. Also plot the cross-section area A, the local velocity v, the specific 
volume V, the absolute temperature T, and the local Mach number M with respect to 
pressure along the nozzle. 

SOLUTION. Assume that the operation of this rocket is at an optimum for expansion to 
sea-level pressure p 3 = 0.1013 MPa so thal the ratio pi/pi = 0.049 in Eq. 3-16. From 
this equation, the effective exhaust velocity is calculated to be 1827 m/sec since here 
V 2 = c. Hence, 



I = c/go = 1827/9.81 = 186 sec and 
F = mc = 1.0 X 1827 = 1827 N 

In order to graph the desired variables as a function of pressure along the nozzle, we 
have to express them in analytically useful ways. In this chapter, we denote the noz- 
zle axial location x as either the chamber (location 1) or the throat (location t) and keep 
the axial location y as a variable which spans the range 1^2 (see Fig. 2-1). The initial 
specific volume Vi is calculated from the equation of state of a perfect gas (Eq. 3-4), 

Pi = RTi/pi = 345.7 X 2222/(2.068 x 10®) = 0.3714 m^kg 

Now we can write equations for the specific volume and the temperature (Eq. 3-7) as 

V, = Pi(pi/p,)'/*^ =0.3714(2.068/p,)°’®® (m^/kg) 

Ty = ri(p,/pi)<*-‘>/*^ = 2222(py/2.06Sf^^^ (K) 



3.3. ISENTROPIC FLOW THROUGH NOZZLES 57 



These can be plotted by inserting py values between 2.068 and 0.1013 in MPa. The 
calculations for the velocity follow from Eq. 3-16 (the derivations have been omitted in 
order to focus on the plotting and are left up to the interested reader), 

Vy = 2580[1 - (Pj,/ 2.068)“-^^']‘''2 (m/sec) 

The cross-sectional area is found from Eq. 3-3 and the relations above: 

Aj, = mVy/Vy = (1.0 X 0.3714/2580) x (2.068/p3,)°™ 

X [1 - (Pj,/2.068)“-^^']-'/2 

= 1.44 X lO-'^ X (2.068 /p 3,)°-’®[1 - (py/2.068)°-^^']“'''^ (m^) 
Einally the Mach number is obtained using Eq. 3-11 and the relations above: 

My = Vy/[kRTy]^''^ = [2580/(1.30 x 345.7 x 2222)'/^] 

X [1 - (Pj,/2.068)“^^]'/^(p,/2.068)“°-2^‘ 

= 2.582[1 - (p3,/2.068)°-^^‘]'/^(Pj,/2.068)“°-^^‘ 

With the equations indicated above and graphing software such as available in MATLAB 
or MAPLE, the desired plots may be obtained. Figure 3-3 shows semiquantitative plots 
of several parameter variations and should be used for comparison. Note that in this figure 
the pressures are shown as decreasing to the right, representing a flow from left to right 
(also its units are in psia where the throat pressure is 164.4 psia and the exit pressure 
would be 14.7 psia). 

A number of interesting deductions can be made from this example. Very high 
gas velocities (over 1 km/sec) can be obtained in rocket nozzles. The temperature 
drop of the combustion gases flowing through a rocket nozzle is appreciable. In 
the example given the temperature changed 1 1 15°C in a relatively short distance. 
This should not be surprising, for the increase in the kinetic energy of the gases is 
derived from a decrease of the enthalpy, which in turn is roughly proportional to 
the decrease in temperature. Because the exhaust gases are still very hot (1107 K) 
when leaving the nozzle, they contain considerable thermal energy not available 
for conversion into kinetic energy of the jet. 



Nozzle Flow and Throat Condition 

The required nozzle area decreases to a minimum (at 1.130 MPa or 164 psi pres- 
sure in the previous example) and then increases again. Nozzles of this type (often 
called De Laval nozzles after their inventor) consist of a convergent section fol- 
lowed by a divergent section. From the continuity equation, the area is inversely 
proportional to the ratio v/V . This quantity has also been plotted in Fig. 3-3. 
There is a maximum in the curve of v/V because at first the velocity increases 



58 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Pressure, mPa 



Pressure, mPa 




Nozzle inlet 



Nozzle inlet 



FIGURE 3-3. Typical variation of cross-sectional area, temperature, specific volume, 
and velocity with pressure in a rocket nozzle (from Example 3-2). 



at a greater rate than the specific volume; however, in the divergent section, the 
specific volume increases at a greater rate. 

The minimum nozzle area is called the throat area. The ratio of the nozzle 
exit area to the nozzle throat area A, is called the nozzle expansion area ratio 
and is designated by the Greek letter e. It is an important nozzle parameter: 



c = Ml At 



(3-19) 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 59 



The maximum gas flow per unit area occurs at the throat where there is a 
unique gas pressure ratio which is only a function of the ratio of specific heats 
k. This pressure ratio is found by setting M = 1 in Eq. 3-13: 



The throat pressure pt for which the isentropic mass flow rate is a maximum 
is called the critical pressure. Typical values of this critical pressure ratio range 
between 0.53 and 0.57 of the inlet pressure. The flow through a specified rocket 
nozzle with a given inlet condition is less than the maximum if the pressure 
ratio is larger than that given by Eq. 3-20. However, note that this ratio is not 
that across the entire nozzle and that the maximum flow or choking condition 
(explained below) is always established internally at the throat and not at the 
exit plane. The nozzle inlet pressure is very close to the chamber stagnation 
pressure, except in narrow combustion chambers where there is an appreciable 
drop in pressure from the injector region to the nozzle entrance region. This is 
discussed in Section 3.5. At the point of critical pressure, namely the throat, the 
Mach number is one and the values of the specific volume and temperature can 
be obtained from Eqs. 3-7 and 3-12: 



In Eq. 3-22 the nozzle inlet temperature Ti is very close to the combustion 
temperature and hence close to the nozzle flow stagnation temperature Tq. At the 
critical point there is only a mild change of these properties. Take, for example, 
a gas with k = 1.2; the critical pressure ratio is about 0.56 (which means that p, 
equals almost half of the chamber pressure pi ; the temperature drops only slightly 
{Tt = 0.91Ti), and the specific volume expands by over 60% {Vt = 1.61Vi). 
From Eqs. 3-15, 3-20, and 3-22, the critical or throat velocity u, is obtained: 



The first version of this equation permits the throat velocity to be calculated 
directly from the nozzle inlet conditions without any of the throat conditions 
being known. At the nozzle throat the critical velocity is clearly also the sonic 
velocity. The flow in the chamber is subsonic and downstream of the nozzle throat 
it is supersonic. The divergent portion of the nozzle permits further decreases in 
pressure and increases in velocity under supersonic conditions. If the nozzle is cut 
off at the throat section, the exit gas velocity is sonic and the flow rate remains 
a maximum. The sonic and supersonic flow condition can be attained only if the 
critical pressure prevails at the throat, that is, if pi/pi is equal to or less than 
the quantity defined by Eq. 3-20. There are, therefore, three different types of 
nozzles: subsonic, sonic, and supersonic, and these are described in Table 3-1. 



P,!p, = [2/{k + l)]'=/(^-D 



(3-20) 



V, = Vi[(k + l)/2]'/(^-^) 
T, = 2Ti/(k + 1) 



(3-21) 

(3-22) 




(3-23) 



60 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



TABLE 3-1. Nozzle Types 

Subsonic Sonic Supersonic 



Throat 

velocity 

Exit 

velocity 

Mach 

number 

Pressure 

ratio 



Shape 



vi < at Vt = at Vt = at 

V2 < a2 V2 = Vt V2 > Vt 

M 2 <1 M2=M, = 1.0 M 2 > 1 




The supersonic nozzle is the one used for rockets. It achieves a high degree 
of conversion of enthalpy to kinetic energy. The ratio between the inlet and exit 
pressures in all rockets is sufficiently large to induce supersonic flow. Only if 
the absolute chamber pressure drops below approximately 1.78 atm will there be 
subsonic flow in the divergent portion of the nozzle during sea-level operation. 
This condition occurs for a very short time during the start and stop transients. 

The velocity of sound is equal to the propagation speed of an elastic pres- 
sure wave within the medium, sound being an infinitesimal pressure wave. If, 
therefore, sonic velocity is reached at any point within a steady flow system, it is 
impossible for a pressure disturbance to travel past the location of sonic or super- 
sonic flow. Thus, any partial obstruction or disturbance of the flow downstream 
of the nozzle throat with sonic flow has no influence on the throat or upstream 
of it, provided that the disturbance does not raise the downstream pressure above 
its critical value. It is not possible to increase the throat velocity or the flow 
rate in the nozzle by further lowering the exit pressure or even evacuating the 
exhaust section. This important condition is often described as choking the flow. 
It is always established at the throat and not the nozzle exit plane. Choked flow 
through the critical section of a supersonic nozzle may be derived from Eqs. 
3-3, 3-21, and 3-23. It is equal to the mass flow at any section within the 
nozzle. 



m = 



AtVt 

V, 



= AtPik 



y/[2/{k + l)]«=+l)/dr-l) 

VMtY 



(3-24) 



The mass flow through a rocket nozzle is therefore proportional to the throat 
area A, and the chamber (stagnation) pressure pp, it is also inversely proportional 
to the square root of T /9J1 and a function of the gas properties. For a supersonic 
nozzle the ratio between the throat and any downstream area at which a pressure 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 61 



Py prevails can be expressed as a function of the pressure ratio and the ratio of 
specific heats, by using Eqs. 3-4, 3-16, 3-21, and 3-23, as follows: 



^ _ VtVy 
Ay VyVt 





1/L 



k + \ 
k - 1 



1 - 1 ^ 
Pi 



(k-iyk- 



(3-25) 



When Py = p 2 , then Ay j A, = A 2 IA, = e in Eq. 3-25. Eor low-altitude operation 
(sea level to about 10,000 m) the nozzle area ratios are typically between 3 and 25, 
depending on chamber pressure, propellant combinations, and vehicle envelope 
constraints. Eor high-altitude (100 km or higher) area ratios are typically between 
40 and 200, but there have been some as high as 400. Similarly, an expression for 
the ratio of the velocity at any point downstream of the throat with the pressure 
Py, and the throat velocity may be written from Eqs. 3-15 and 3-23: 



Vy _ 


k + \ 


1 

7 

1 

1 




k - 1 


\pi/ 



(3-26) 



These equations permit the direct determination of the velocity ratio or the 
area ratio for any given pressure ratio, and vice versa, in ideal rocket nozzles. 
They are plotted in Eigs. 3-4 and 3-5, and these plots allow the determination 
of the pressure ratios given the area or velocity ratios. When py = p 2 , Eq. 3-26 
describes the velocity ratio between the nozzle exit area and the throat section. 
When the exit pressure coincides with the atmospheric pressure (p 2 = P 3 , see 
Eig. 2-1), these equations apply for optimum nozzle expansion. Eor rockets 
that operate at high altitudes, not too much additional exhaust velocity can be 
gained by increasing the area ratio above 1000. In addition, design difficulties 
and a heavy inert nozzle mass make applications above area ratios of about 350 
marginal. 

Appendix 2 is a table of several properties of the earth’s atmosphere with 
agreed-upon standard values. It gives ambient pressure for different altitudes. 
These properties can vary somewhat from day to day (primarily because of solar 
activity) and between hemispheres. Eor example, the density of the atmosphere 
at altitudes between 200 and 3000 km can change by more than an order of 
magnitude, affecting satellite drag. 



Example 3-3. Design an ideal nozzle for a rocket that operates at 25 km altitude and 
delivers 5000 N thrust with a chamber pressure of 2.039 MPa and a chamber temperature 
of 2800 K. Assuming that k = \.2Q and R = 360 J/kg-K, determine the throat area, exit 
area, throat temperature, and exit velocity. 



SOLUTION. At 25 km altitude the atmospheric pressure equals 2.549 kPa (in Appendix 
2 the ratio is 0.025158, which must be multiplied by the pressure at sea level, 0.1013 
MPa). The controlling pressure ratio is 



P 2 /P 1 = 0.002549/2.039 = 0.00125 = 1/800 



62 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 




10 15 20 25 30 40 50 60 80 100 150 200 300 



P1 



FIGURE 3-4. Area and velocity ratios as function of pressure ratio for the diverging 
section of a supersonic nozzle. 



The area ratio can now he obtained directly from Eq. 3-25 (or Fig. 3-5) as 

A, a + U + i 

Ai \ 2 } U/ ^k-l 

= (1.10)^(0.00125)°*^Vll-0[l - (0.00125)‘’ 'S’] 

= 1.67 X 10“^ = 1/60 



(k-i)/k 



The temperature at the throat may be found directly from Eq. 3-22: 
T, = 2Ti/(k + 1) = 2 X 2800/2.2 = 2545 K 
The ideal exit velocity is found from Eq. 3-15 as 



T2 



N 



2k 

k - 1 



RTi 



1-1^ 
Pi 






yi.21 X 10’[1 - (0.00125)01®’] 



= 2851 m/sec 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 63 




300 500 1000 2000 3000 5000 10,000 

Pi 



FIGURE 3-5. Continuation of prior figure of area ratios and velocity ratios, but for 
higher pressure ratios in a supersonic nozzle. 



Next we need to find the throat area, which can be found via the mass flow rate, Eq. 3-24, 
A, = mVt/vt. To find the mass flow rate explicitly we note that V 2 = c (Eq. 2-16) and 
from Eq. 2-17 



m = Fjc = 5000/2851 = 1.754 kg/sec 
A = ” / 

' Pi Y /t[2/(fe + l)](*+i)/(^-0 



1.754 / 360 X 2800 

2.039 X lO^Y 1.2[2/2.2]ii 



= 13.32 cm^ 



And thus the exit area becomes A 2 = 60 x 13.32 = 799 cm^. The designer would next 
have to modify the area ratio of the nozzle depending on the actual configuration cho- 
sen and to choose materials together with cooling methods that accommodate the high 
chamber and throat temperatures. 



64 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Thrust and Thrust Coefficient 

The efflux of the propellant gases or the momentum flowing out causes the 
thrust or reaction force on the rocket structure. Because the flow is supersonic, 
the pressure at the exit plane of the nozzle may he different from the ambient 
pressure, and the pressure thrust component adds to the momentum thrust as 
given by Eq. 2-14, which is repeated here: 

F = mV2 + (P2 -P3)M 

The maximum thrust for any given nozzle operation is found in a vacuum 
where = 0. Between sea level and the vacuum of space, Eq. 2-14 gives the 
variation of thrust with altitude, using the properties of the atmosphere such as 
those listed in Appendix 2. Eigure 2-2 shows a typical variation of thrust with 
altitude. To modify values calculated for optimum operating conditions (p 2 = Pi) 
for given values of pi, k, and A 2 /A,, the following expressions may be used. Eor 
the thrust. 



F = Fopi+piAf 




Pi) A, 



For the specific impulse, using Eqs. 2-5, 2-18, and 2-14, 



c € ( P 2 

Is = (/Jopt + - 

go \Pl 




(3-27) 



(3-28) 



If, for example, the specific impulse for a new exit pressure p 2 corresponding to 
a new area ratio A 2 /A, is to be calculated, the above relations may be used. 

Equation 2-14 can be expanded by modifying it and substituting V 2 ,v,, and 
Vt from Eqs. 3-16, 3-21, and 3-23: 



A,V,V2 

F = — h (P2 -Pi)A2 



= A,pi 






2k^ 

k — 1 V A: + 1 



(k+i)/(k-i) 



(k-D/k- 



+ (P2- Pi) A 2 

(3-29) 



The first version of this equation is general and applies to all rockets, the sec- 
ond form applies to an ideal rocket with k being constant throughout the expan- 
sion process. This equation shows that the thrust is proportional to the throat area 
At and the chamber pressure (or the nozzle inlet pressure) pi and is a function 
of the pressure ratio across the nozzle pi/p 2 , the specific heat ratio k, and of the 
pressure thrust. It is called the ideal thrust equation. The thrust coefficient Cf is 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 65 



defined as the thrust divided by the chamber pressure p\ and the throat area A,. 
Equations 2-14, 3-21, and 3-16 then give this relation: 



^ U2A2 ^ P2A2 P3A2 
PiA,V2 Pi At piA, 






2k^ 



k-l\k+l 



(k+\)/(k-i) 



i-r- 



(k-i)/k- 



+ — — —— (3-30) 
Pi At 



The thrust coefficient Cp is dimensionless. It is a key parameter for analysis 
and is a function of gas property k, the nozzle area ratio e, and the pressure 
ratio across the nozzle pi/pi, but independent of chamber temperature. For any 
hxed pressure ratio pijps, the thrust coefficient Cp and the thrust F have a peak 
when p 2 = Pi- This peak value is known as the optimum thrust coefficient and 
is an important criterion in nozzle design considerations. The use of the thrust 
coefficient permits a simplihcation to Eq. 3-29: 



F = CpAtpi (3-31) 

Equation 3-31 can be solved for Cp and provides the relation for determining 
the thrust coefficient experimentally from measured values of chamber pres- 
sure, throat diameter, and thrust. Even though the thrust coefficient is a function 
of chamber pressure, it is not simply proportional to pi, as can be seen from 
Eq. 3-30. However, it is directly proportional to throat area. The thrust coef- 
ficient can be thought of as representing the amplification of thrust due to the 
gas expanding in the supersonic nozzle as compared to the thrust that would 
be exerted if the chamber pressure acted over the throat area only. The thrust 
coefficient has values ranging from about 0.8 to 1.9. It is a convenient parameter 
for seeing the effects of chamber pressure or altitude variations in a given nozzle 
configuration, or to correct sea-level results for flight altitude conditions. 

Figure 3-6 shows the variation of the optimum expansion thrust coefficient 
(p 2 = pf) for different pressure ratios p\lp 2 , values of k, and area ratio e. The 
complete thrust coefficient is plotted in Figs. 3-7 and 3-8 as a function of 
pressure ratio p\/p 2 and area ratio for k = 1.20 and 1.30. These two sets of curves 
are useful in solving various nozzle problems for they permit the evaluation of 
under- and over-expanded nozzle operation, as explained below. The values given 
in these figures are ideal and do not consider such losses as divergence, friction, 
or internal expansion waves. 

When p\fp 2 , becomes very large (e.g., expansion into near vacuum), then the 
thrust coefficient approaches an asymptotic maximum as shown in Figs. 3-7 and 
3-8. These hgures also give values of for any mismatched nozzle (p 2 Ps), 
provided the nozzle is flowing full at all times, that is, the working fluid does 
not separate or break away from the walls. Flow separation is discussed later in 
this section. 




o 

o 

o 

o 



o 

o 

o 

m 



o 

o 

o 

CM 



o 

o 

o 



o 

o 

m 




qT 



o 

o 

CM 



o 

o 



o 

in 



o 

CM 



O 



66 



FIGURE 3-6. Thrust coefficient Cp as a function of pressure ratio, nozzle area ratio, and specific heat ratio for optimum expansion conditions 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 67 




Area ratio e = Ag/A, 

FIGURE 3-7. Thrust coefficient Cf versus nozzle area ratio for k = 1.20. 




FIGURE 3-8. Thrust coefficient Cf versus nozzle area ratio for k = 1.30. 



68 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Characteristic Velocity and Specific Impulse 

The characteristic velocity c* was defined by Eq. 2-18. From Eqs. 3-24 and 
3-31 it can be shown that 

^ ^ ^ JL = ^/kRTl 

^ m Cf Cf ky/[2l(k + l)](*:+l)/d^-l) 

It is basically a function of the propellant characteristics and combustion chamber 
design; it is independent of nozzle characteristics. Thus, it can be used as a figure 
of merit in comparing propellant combinations and combustion chamber designs. 
The first version of this equation is general and allows the determination of c* 
from experimental data of m,pi, and A,. The last version gives the maximum 
value of c* as a function of gas properties, namely k, the chamber temperature, 
and the molecular mass 971, as determined from the theory in Chapter 5. Some 
values of c* are shown in Tables 5-4 and 5-5. 

The term c*-efficiency is sometimes used to express the degree of completion 
of the energy release and the creation of high-temperature, high-pressure gas 
in the chamber. It is the ratio of the actual value of c*, as determined from 
measurements (the first part of Eq. 3-32), and the theoretical value (last part of 
Eq. 3-32), and typically has a value between 92 and 99.5%. 

Using Eqs. 3-31 and 3-32, the thrust itself may now be expressed as the 
mass flow rate times a function of the combustion chamber (c*) times a function 
of the nozzle expansion Cf), 



F = CFihc* (3-33) 

Some authors use a term called the discharge coefficient Co , which is merely 
the reciprocal of c*. Both Co and the characteristic exhaust velocity c* are used 
primarily with chemical rocket propulsion systems. 

The influence of variations in the specific heat ratio k on various parame- 
ters (such as c,c*A 2 lAt,V 2 lvu or f) is not as large as the changes in chamber 
temperature, pressure ratio, or molecular mass. Nevertheless, it is a noticeable 
factor, as can be seen by examining Figs. 3-2 and 3-4 to 3-8. The value of k is 
1.67 for monatomic gases such as helium and argon, 1.4 for cold diatomic gases 
such as hydrogen, oxygen, and nitrogen, and for triatomic and beyond it varies 
between 1.1 and 1.3 (methane is 1.11 and ammonia and carbon dioxide 1.33). 
In general, the more complex the molecule the lower the value of k\ this is also 
true for molecules at high temperatures when their vibrational modes have been 
activated. The average values of k and 971 for typical rocket exhaust gases with 
several constituents depend strongly on the composition of the products of com- 
bustion (chemical constituents and concentrations), as explained in Chapter 5. 
Values of k and 971 are given in Tables 5-4, 5-5, and 5-6. 

Example 3-4. What is the percentage variation in thrust between sea level and 10 km 
for a launch vehicle whose rocket operates with a chamber pressure of 20 atm and has 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 69 



an expansion area ratio of 6? (Use k = 1.30.) Also, at what altitude is the performance 
of this rocket optimum? 

SOLUTION. According to Eq. 3-33, the only component of the thrust that depends 
on the ambient pressure is Cp, the thrust coefficient. This coefficient can be found from 
Eq. 3-30 or from Fig. 3-8 by following a vertical line corresponding to AijAt = 6.0. We 
first have to find the ratio pi/ps at sea level and at 10 km. Using values from Appendix 2 
and solving Eq. 3-25 for P 2 /P 3 = 0.0198(p2 = 0.396 atm). 

Sea level: pi/p 3 = 20/1.0 = 20 Cf = 1.33 

10 km altitude: pi/p 3 = 20/0.26151 = 76.5 Cp = 1.56 

Thrust increase = (1.56 - 1.33)/1.33 = 17.3% 

Note that performance at optimum nozzle expansion (p 2 = P 3 ) this nozzle delivers a 
thrust that is somewhat less than the maximum. From Fig. 3-6 we find that for e = 6 
the optimum Cp = 1.52, which corresponds to 7.2 km elevation, whereas the maximum 
value is Cp = 1.63 in a vacuum. The thrust coefficient could be somewhat higher for 
larger area ratios. 



Under- and Over-Expanded Nozzles 

An under-expanded nozzle discharges the fluid at an exit pressure greater than 
the external pressure because the exit area is too small for an optimum area ratio. 
The expansion of the fluid is therefore incomplete within the nozzle, and further 
expansion will take place outside of the nozzle exit. The nozzle exit pressure is 
higher than the local atmospheric pressure. 

In an over-expanded nozzle the fluid exists at lower pressure than the 
atmosphere as it has an exit area too large for optimum. The phenomenon of 
over-expansion for a supersonic nozzle is shown in Fig. 3-9, with typical 
pressure measurements of superheated steam along the nozzle axis and different 
back pressures or pressure ratios. Curve AB shows the variation of pressure 
with the optimum back pressure corresponding to the nozzle area ratio. Curves 
AC and AD show the variation of pressure along the axis for increasingly 
higher external pressures. The expansion within the nozzle proceeds normally 
for the initial portion of the nozzle. At point I on curve AD, for example, the 
pressure is lower than the exit pressure and a sudden rise in pressure takes place 
that is accompanied by the separation of the flow from the walls (separation is 
described later). 

The nonideal behavior of nozzles is strongly influenced by the presence of 
compression waves or shock waves inside the diverging nozzle section, which are 
strong compression discontinuities and exist only in supersonic flow. The sudden 
pressure rise in the curve ID is such a compression wave. Expansion waves, also 
strictly supersonic phenomena, match the flow from a nozzle exit to lower ambi- 
ent pressures. Compression and expansion waves are described in Chapter 20. 

The different possible flow conditions in a supersonic nozzle are as follows: 

1. When the external pressure p 3 is below the nozzle exit pressure p 2 , the 
nozzle will flow full but will have external expansion waves at its exit (i.e., 



70 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Divergence 




FIGURE 3-9. Distribution of pressures in a converging-diverging nozzle for different 
flow conditions. Inlet pressure is the same, but exit pressure changes. Based on experi- 
mental data from A. Stodala. 



under-expansion). The expansion of the gas inside the nozzle is incomplete 
and the values of Cf and 4 will be less than at optimum expansion. 

2. For external pressures slightly higher than the nozzle exit pressure p 2 , 
the nozzle will continue to flow full. This occurs until p 2 reaches a value 
between about 10 and 40% of p^. The expansion is somewhat inefficient 
and Cf and 4 will have lower values than an optimum nozzle would have. 
Shock waves will exist outside the nozzle exit section. 

3. For higher external pressures, separation of the flow will begin to take place 
inside the divergent portion of the nozzle. The diameter of the supersonic jet 
will be smaller than the nozzle exit diameter. With steady flow, separation 
is typically axially symmetric. Figures 3-10 and 3-11 show diagrams of 
separated flows. The axial location of the separation plane depends on 
the local pressure and the wall contour. The point of separation travels 
downstream with decreasing external pressure. At the nozzle exit the flow 
in the center portion remains supersonic but is surrounded by an annular- 
shaped section of subsonic flow. There is a discontinuity at the separation 
location and the thrust is reduced, compared to a nozzle that would have 



3.3. ISENTROPIC FLOW THROUGH NOZZLES 71 




Altitude, m 

FIGURE 3-10. Thrust coefficient Cp for two nozzles with different area ratios. One has 
jet separation below about 7000 m altitude. The fully expanded exhaust plume at higher 
altitudes is not shown in the sketch. 



been cut off at the separation plane. Shock waves exist outside the nozzle 
in the external plume. 

4. For nozzles in which the exit pressure is just below the value of the inlet 
pressure, the pressure ratio is below the critical pressure ratio (as defined 
by Eq. 3-20) and subsonic flow prevails throughout the entire nozzle. This 
condition occurs normally in rocket nozzles for a short time during the start 
and stop transients. 

The method for estimating pressure at the location of the separation plane 
inside the diverging section of a supersonic nozzle has usually been empirical. 
Reference 3-4 describes a variety of nozzles, their behavior, and methods used 
to estimate the location and the pressure at separation. Actual values of pressure 
for the over-expanded and under-expanded regimes described above are functions 
of the specific heat ratio and the area ratio (see Ref. 3-1). 

The axial thrust direction is not usually altered by separation because a steady 
flow usually separates uniformly over a cross section in a divergent nozzle cone 
of conventional rocket design. During transients, such as start and stop, the sep- 
aration may not be axially symmetric and may cause momentary but large side 
forces on the nozzle. During a normal sea-level transient of a large rocket nozzle 



72 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



During flight During sea level static tests 




FIGURE 3-11. Simplified sketches of exhaust gas behavior of three typical rocket noz- 
zles for a three-stage launch vehicle. The first vehicle stage has the biggest chamber and 
the highest thrust but the lowest nozzle area ratio, and the top or third stage usually has 
the lower thrust but the highest nozzle area ratio. 



(before the chamber pressure reaches its full value), some momentary flow oscil- 
lations and nonsymmetric separation of the jet can occur during over-expanded 
flow operation. The magnitude and direction of transient side forces can change 
rapidly and erratically. The resulting side forces can be large and have caused 
failures of nozzle exit cone structures and thrust vector control gimbal actua- 
tors. References 3-4, 3-5, and 3-6 discuss techniques for estimating these side 
forces. 

When the flow separates, as it does in a highly over-expanded nozzle, the 
thrust coefficient Cp can be estimated if the point of separation in the nozzle is 
known. Thus, Cf can be determined for an equivalent smaller nozzle with an 
exit area equal to that at the point of separation. The effect of separation is to 
increase the thrust and the thrust coefficient over the value that they would have 
if separation had not occurred. Thus, with separated gas flow, a nozzle designed 
for high altitude (large value of e) would have a larger thrust at sea level than 
expected, but not as good as an optimum nozzle; in this case separation may 
actually be desirable. With separated flow a large and usually heavy portion of 
the nozzle is not utilized and the nozzle is bulkier and longer than necessary. The 
added engine weight and size decrease flight performance. Designers therefore 
select an area ratio that will not cause separation. 



3.3. ISENTROPIC FLOWTHROUGH NOZZLES 73 



Because of uneven flow separation and potentially destructive side loads, sea- 
level static tests of an upper stage or a space propulsion system with a high area 
ratio over-expanded nozzle are usually avoided; instead, a sea-level test nozzle 
with a much smaller area ratio is substituted. However, actual and simulated alti- 
tude testing (in an altitude test facility similar to the one described in Chapter 21) 
would be done with a nozzle having the correct large area ratio. The ideal solu- 
tion that avoids separation at low altitudes and has high values of Cp at high 
altitudes is a nozzle that changes area ratio in flight. This is discussed at the end 
of this section. 

For most applications, the rocket system has to operate over a range of alti- 
tudes; for a fixed chamber pressure this implies a range of nozzle pressure ratios. 
The condition of optimum expansion (p 2 = ps) occurs only at one altitude, and 
a nozzle with a fixed area ratio is therefore operating much of the time at either 
over-expanded or under-expanded conditions. The best nozzle for such an appli- 
cation is not necessarily one that gives optimum nozzle gas expansion, but one 
that gives the largest vehicle flight performance (say, total impulse, or specific 
impulse, or range, or payload); it can often be related to a time average over the 
powered flight trajectory. 

Example 3-5. Use the data from Example 3-4 (pi = 20 atm, fc = 1.30) and consider 
both the area ratio 6 = 6.0 and e = 15.0. Compare the performance of these two nozzles 
as a function of altitude by plotting Cp up to 50 km. Assume no shocks inside the nozzle. 

SOLUTION. The procedures are the same as those in Example 3-4 and the resulting 
plot is shown in Fig. 3-10. The values of Cp can be obtained by following vertical lines 
for e = 6.0 and 15.0 in Fig. 3-8 corresponding to different pressure ratios at increasing 
altitudes. Alternatively, Eq. 3-30 may be used for greater accuracy. The lower area ratio 
gives a better performance at the lower altitudes but above about 10 km the larger area 
ratio gives a higher Cp. Also noted in Fig. 3-10 by the dotted line is what a continuously 
variable nozzle area ratio could accomplish by matching p 2 to pa at all altitudes. 

The optimum pressure ratio always occurs when pi/pa = p\lpi and for the e = 15.0 
case, from Fig. 3-6 or 3-8, this value is about 180; pa = 20/180 = 0.111 atm, which 
corresponds to about 15.5 km altitude. Below this altitude the nozzle is over-expanded. 
At sea level, where pi/pa =20, separation could occur as indicated in Fig. 3-8. For 
similarly shaped nozzles, it is estimated that sea-level separation begins to take place at 
a cross section where the local pressure is between 10 and 40% of pa or below 0.4 atm. 
Upon separation, this nozzle would not flow full downstream of an area ratio of about 6 
or 7 and the gas jet would occupy only the central portion of the nozzle exit. See Figs. 
3-10 and 3-11. Weak shock waves and jet contraction would then raise the exhaust jet’s 
pressure to match the value of the surrounding atmosphere. If the jet does not separate, 
it would reach an exit pressure of 0. 1 1 atm, but this is often an unstable condition. As 
the vehicle gains altitude, the separation plane would gradually move downstream until, 
at an altitude of about 700 m, the exhaust gases would occupy the full diverging section 
of the nozzle area. 

The values of Cp can be obtained by following a vertical line for e = 15 and e = 6 
in Fig. 3-8 for different pressure ratios, which correspond to different altitudes. Alterna- 
tively, Eq. 3-30 can be used for better accuracy. Results are similar to those plotted in 
Fig. 3-10. The lower area ratio of 6 gives a higher Cp at low altitudes but is inferior to 



74 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



the area ratio of 15 at the higher altitudes. A nozzle area ratio larger than 15 would yield 
a higher Cp at the higher altitudes. 

Figure 3-11 shows a comparison of altitude and sea-level behavior of three 
nozzles and their plumes at different area ratios for a typical three-stage satellite 
launch vehicle. When fired at sea-level conditions, the nozzle of the third stage 
with the highest area ratio will experience flow separation and suffer a major 
performance loss; the second stage will flow full but the external plume will 
contract; since p 2 < Pi, there is a loss in /j and F . There is no effect on the first 
stage nozzle. 

Example 3-6. A rocket engine test near sea level gives the following data: thrust F = 
53,000 Ibf, propellant mass flow rate m = 208 Ibm/sec, nozzle exit area ratio A^jAt = 
10.0, actual local atmospheric pressure at test station = 13.8 psia, and chamber pressure 
p\ = 620 psia. The test engineer also knows that for the same flow rate (and mixture 
ratio) the theoretical specific impulse is 289 sec at the standard reference conditions of 
Pi = 1000 psia and p 3 = 14.7 psia, and that k = 1.20. For many propellants we may 
assume that the combustion temperature and k do not vary significantly with chamber 
pressure. Compare the test performance of this rocket with its equivalent at sea level, 
standard, and vacuum performance. 

SOLUTION. The pressure ratio for the test condition is 620/13.8 = 44.93; if the test 
would have been conducted at 14.7 psia, this pressure ratio would have been 620/14.7 = 
42.18; for the standard reference conditions the pressure ratio is 1000/14.7 = 68.03. 

Since this nozzle is operating supersonically, we find, from Eqs. 2-5, 3-24, and 
3-31, that specific impulse ratios depend only on ratios of the thrust coefficient, that is, 
(Di, = (Is)al{CF)h / (CF)a]- From Fig. 3-7 or from Eq. 3-30 one obtains Cp = 1.48 for 
the test condition with e = 10,k = 1.20, and pi/ps = 44.9. The corresponding specific 
impulse is, from Eq. 2-5, L = 53000/208 = 255 sec. 

The following table shows all the results: 





Test 

Results 


Test for 
Sea Level 


Standard 

Condition 


Vacuum 

Condition 


Pi (psia) 


620 


620 


1,000 


620 


P 3 (psia) 


13.8 


14.7 


14.7 


0 


Pl/P3 


44.93 


42.18 


68.03 


OO 


F (Ibf) 


53,000 


52,494 


55,744 


60,741 


h (sec) 


255 


252 


268 


292 


Cf 


1.52 


1.50 


1.60 


1.74 



Figures 3-10 and 3-11 suggest that an ideal design for an ascending (e.g., 
launch) rocket vehicle would have a “rubber-like” diverging section that could be 
lengthened and enlarged so that the nozzle exit area could be made larger as the 
ambient pressure is reduced. The design would then allow the rocket vehicle to 
attain its maximum performance at all altitudes as it ascends. As yet we have not 
achieved a simple mechanical hardware design with this full altitude compen- 
sation similar to “stretching rubber.” However, there are a number of practical 



3.4. NOZZLE CONFIGURATIONS 75 



TABLE 3-2. Estimated Losses for Small-Diameter Chambers 



Chamber-to-Throat 
Area Ratio 


Throat 
Pressure (%) 


Thrust 

Reduction (%) 


Specific Impulse 
Reduction (%) 


oo 


100 


0 


0 


3.5 


99 


1.5 


0.31 


2.0 


96 


5.0 


0.55 


1.0 


81 


19.5 


1.34 



k = 1.20; Pi /p2 = 1000. 



nozzle configurations that can be used to alter the flow shape with altitude and 
obtain maximum performance at different altitudes. They are discussed in the 
next section. 

Influence of Chamber Geometry 

When the chamber has a cross section that is larger than about four times the 
throat area (Ai/A, > 4), the chamber velocity ui, can be neglected, as was men- 
tioned in explaining Eqs. 3-15 and 3-16. However, vehicle space or weight 
constraints often require smaller thrust chamber areas for liquid propellant engines 
and grain design considerations lead to small void volumes or small perforations 
or port areas for solid propellant motors. Then iq can no longer be neglected as 
a contribution to the performance. The gases in the chamber expand as heat is 
being added. The energy necessary to accelerate these expanding gases within 
the chamber will also cause a pressure drop and an additional energy loss. This 
acceleration process in the chamber is adiabatic (no heat transfer) but not isen- 
tropic. This loss is a maximum when the chamber diameter is equal to the nozzle 
diameter, which means that there is no converging nozzle section. This has been 
called a throatless rocket motor and has been used in a few tactical missile booster 
applications, where there was a premium on minimum inert mass and length. The 
flight performance improvement due to inert mass savings supposedly outweighs 
the nozzle performance loss of a throatless motor. Table 3-2 lists some of the 
performance penalties for three chamber area ratios. 

Because of this pressure drop within narrow chambers, the chamber pressure 
is lower at the nozzle entrance than it would be if Ai/A, had been larger. This 
causes a small loss in thrust and specific impulse. The theory of this loss is given 
in Ref. 3-7 and some results are listed in Table 3-2. 



3.4. NOZZLE CONFIGURATIONS 

A number of different proven nozzle configurations are available today. This 
section describes their geometries and performance. Other chapters (6, 8, 12, 
15, and 18) discuss their materials, heat transfer, or application and mention 



76 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



their requirements, design, construction, and thrust vector control. Nozzles and 
chambers are usually of circular cross section and have a converging section, a 
throat at the narrowest location (minimum cross section), and a diverging section. 
Nozzles can be seen in Figs. 1-4, 1-5, 1-8, 2-1, 3-11, 3-12, 3-14, 11-1, 
12-1 to 12-3, and 15-6 to 15-8. Refs. 3-4 and 3-8 describe many nozzle 
configurations. 

The converging nozzle section between the chamber and the nozzle throat 
has never been critical in achieving high performance. The subsonic flow in this 
section can easily be turned at very low pressure drop and any radius, cone angle, 
wall contour curve, or nozzle inlet shape is usually satisfactory. A few small 
attitude control thrust chambers have had their nozzle at 90° from the combustion 
chamber axis without any performance loss. The throat contour also is not very 
critical to performance, and any radius or other curve is usually acceptable. The 
pressure gradients are high in these two regions and the flow will adhere to the 
walls. The principal difference in the different nozzle configurations is found 
in the diverging supersonic-flow section, as described below. The wall surface 
throughout the nozzle should be smooth and shiny to minimize friction, radiation 
absorption, and convective heat transfer due to surface roughness. Gaps, holes, 
sharp edges, or protrusions must be avoided. 

Six different nozzle configurations are shown in Fig. 3-12 and each will be 
discussed. The first three sketches show conical and bell-shaped nozzles. The 
other three have a center body inside the nozzle and have excellent altitude com- 
pensation. Although these last three have been ground tested, to date none of 
them has flown in a production space launch vehicle. The lengths of several 
nozzle types are compared in Fig. 3-13. The objectives of a good nozzle con- 
figuration are to obtain the highest practical 7^, minimize inert nozzle mass, and 
conserve length (shorter nozzles can reduce vehicle length, vehicle structure, and 
vehicle inert mass). 



Cone- and Bell-Shaped Nozzles 

The conical nozzle is the oldest and perhaps the simplest configuration. It is 
relatively easy to fabricate and is still used today in many small nozzles. A 
theoretical correction factor k can be applied to the nozzle exit momentum of 
an ideal rocket with a conical nozzle exhaust. This factor is the ratio between 
the momentum of the gases in a nozzle with a finite nozzle angle 2a and the 
momentum of an ideal nozzle with all gases flowing in an axial direction: 

k = i(l -t-cosa) (3-34) 

The variation of A with different values of a is shown in Table 3-3 for any 
nozzle that has uniform mass flow per unit exit area. For ideal rockets A = 1.0. 
For a rocket nozzle with a divergence cone angle of 30° (half angle a = 15°), the 
exit momentum and therefore the exhaust velocity will be 98.3% of the velocity 
calculated by Eq. 3- 15b. Note that the correction factor A only applies to the 



Cone Contoured or Contoured or Plug or Plug or Expansion- 

(15° half angle) bell-full length bell shape, aerospike full aerospike, deflection 

shortened length truncated or cut off 




77 



FIGURE 3-12. Simplified diagrams of several different nozzle configurations and their flow effects. 



78 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 




FIGURE 3-13. Length comparison of several types of nozzles. (Taken in part from Ref 
3-9.) 



TABLE 3-3. Nozzle Angle Correction Factor for Conical Nozzles 



Nozzle Cone Divergence Half 
Angle, a (deg) 


Correction Factor, X 


0 


1.0000 


2 


0.9997 


4 


0.9988 


6 


0.9972 


8 


0.9951 


10 


0.9924 


12 


0.9890 


14 


0.9851 


15 


0.9830 


16 


0.9806 


18 


0.9755 


20 


0.9698 


22 


0.9636 


24 


0.9567 



first term (the momentum thrust) in Eqs. 2-14, 3-29, and 3-30 and not to the 
second term (pressure thrust). 

A small nozzle divergence angle causes most of the momentum to be axial 
and thus gives a high specific impulse, but the long nozzle has a penalty in 
rocket propulsion system mass, vehicle mass, and also design complexity. A large 



3.4. NOZZLE CONFIGURATIONS 79 



divergence angle gives short, lightweight designs, but the performance is low. 
There is an optimum conical nozzle shape and length (typically between 12 and 
18° half angle) and it is usually a compromise which depends on the specific 
application and flight path. 

The bell-shaped or contour nozzle (see Figs. 3-12 and 3-13) is probably the 
most common nozzle shape today. It has a high angle expansion section (20 to 
50°) right downstream of the nozzle throat; this is followed by a gradual reversal 
of nozzle contour slope so that at the nozzle exit the divergence angle is small, 
usually less than a 10° half angle. It is possible to go to large divergence angles 
immediately behind the throat (20 to 50°) because the high relative pressure, the 
large pressure gradient, and the rapid expansion of the working fluid do not allow 
separation in this region unless there are discontinuities in the nozzle contour. 
The expansion in the supersonic bell nozzle is more efficient than in a simple 
straight cone of similar area ratio and length, because the wall contour is designed 
to minimize losses, as explained later in this section. For the past several decades 
most of the nozzles have been bell shaped. 

A change of flow direction of a supersonic gas in an expanding wall geometry 
can only be achieved through expansion waves. An expansion wave occurs at a 
thin surface within the flow, where the flow velocity increases and changes its 
flow direction slightly, and where the pressure and temperature drop. These wave 
surfaces are at an oblique angle to the flow. As the gas passes through the throat, 
it undergoes a series of these expansion waves with essentially no loss of energy. 
In the bell-shaped nozzle shown in Fig. 3-14 these expansions occur internally 
in the flow between the throat and the inflection location /; the area is steadily 
increasing like a flare on a trumpet. The contour angle 0, is a maximum at the 
inflection location. Between the inflection point I and the nozzle exit E the flow 
area is still increasing, but at a diminishing rate, allowing further gas expansion 
and additional expansion waves. However, the contour of the nozzle wall is 
different and the change in cross-sectional area per unit length is decreasing. The 
purpose of this last segment of the contoured nozzle is to have a low divergence 
loss as the gas leaves the nozzle exit plane. The angle at the exit 9^ is small, 
usually less than 10°. The difference between 0, and 6e is called the turn-back 
angle. When the gas flow is turned in the opposite direction (between points 
I and E) oblique compression waves will occur. These compression waves are 
thin surfaces where the flow undergoes a mild shock, the flow is turned, and the 
velocity is actually reduced slightly. Each of these multiple compression waves 
causes a small energy loss. By carefully determining the wall contour (by an 
analysis that uses a mathematical tool called the method of characteristics), it is 
possible to balance the oblique expansion waves with the oblique compression 
waves and minimize the energy loss. The analysis leading to the nozzle contour 
is presented in Chapter 20.33 of Ref. 3-3 and also in Refs. 3-8 to 3-11; it is 
based on supersonic aerodynamic flow, the method of characteristics (Ref. 3-1), 
and the properties of the expanding gas. Most of the rocket organizations have 
computer codes for this analysis. The radius of curvature or the contour shape at 



80 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 






Percent of length of a 15 deg half-angle 
conical nozzle with same area as bell shape 



FIGURE 3-14. Top sketch shows comparison of nozzle inner wall surfaces for a 15° 
conical nozzle, an 80% length bell nozzle, a 60% length bell nozzle, all at an area ratio 
of 25. The lengths are expressed in multiples of the throat radius r,. The middle set of 
curves shows the initial angle 0, and the exit angle 0 ^ for bell nozzles as functions of the 
nozzle area ratio and percent length. The bottom curves show the nozzle losses in terms 
of a correction factor. 



3.4. NOZZLE CONFIGURATIONS 81 



the throat region have an influence on the contour of the diverging bell- shaped 
nozzle section. 

The length of a bell nozzle is usually given a fraction of the length of a ref- 
erence conical nozzle with a 15° half angle. An 80% bell nozzle has a length 
(distance between throat plane and exit plane) that is 20% shorter than a compa- 
rable 15° cone of the same area ratio. Ref. 3-9 shows the original presentation 
by Rao of the method of characteristics applied to shorter bell nozzles. He also 
determined that a parabola was a good approximation for the bell-shaped con- 
tour curve (Ref. 3-3, Section 20.33), and parabolas have actually been used in 
some nozzle designs. The top part of Fig. 3-14 shows that the parabola is tan- 
gent (9j) at the inflection point I and has an exit angle (0g) at point E and a 
length L that has to be corrected for the curve 77. These conditions allow the 
parabola to be determined by simple geometric analysis or geometric drawing. 
A throat approach radius of 1.5r, and a throat expansion radius of 0.4r, were 
used. If somewhat different radii had been used, the results would have been only 
slightly different. The middle set of curves gives the relation between length, area 
ratio, and the two angles of the bell contour. The bottom set of curves gives the 
correction factors, equivalent to the A factor for conical nozzles, which are to be 
applied to the thrust coefficient or the exhaust velocity, provided the nozzles are 
at optimum expansions, that is, p 2 = p^. 

Table 3-4 shows data for parabolas developed from this figure, which allow 
the reader to apply this method and check the results. The table shows two 
shortened bell nozzles and a conical nozzle, each for three area ratios. It can be 
seen that as the length has been decreased, the losses are higher for the shorter 
length and slightly higher for small nozzle area ratios. A 1% improvement in 
the correction factor gives about 1% more specific impulse (or thrust) and this 
difference can be significant in many applications. The reduced length is an 



TABLE 3-4. Data on Several Bell-Shaped Nozzles 



Area Ratio 


10 


25 


50 


Cone (15° Half Angle) 








Length (100%)“ 


8.07 


14.93 


22.66 


Correction factor A 
80% Bell Contour 


0.9829 


0.9829 


0.9829 


Length“ 


6.45 


11.94 


18.12 


Correction factor A 
Approximate half angle at inflection 
point and exit (degrees) 


0.985 


0.987 


0.988 


60% Bell Contour 








Length” 


4.84 


9.96 


13.59 


Correction factor A 


0.961 


0.968 


0.974 


Approximate half angle at inflection 
point and exit (degrees) 


32.5/17 


36/14 


39/18 



The length is given in dimensionless form as a multiple of the throat radius. 



82 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



important benefit, and it is usually reflected in an improvement of the vehicle 
mass ratio. The table and Fig. 3-14 show that bell nozzles (75 to 85% length) 
are just as efficient as or slightly more efficient than a longer 15° conical nozzle 
(100% length) at the same area ratio. For shorter nozzles (below 70% equivalent 
length) the energy losses due to internal oblique shock waves become substantial 
and such short nozzles are not commonly used today. 

For solid propellant rocket motor exhausts with small solid particles in the gas 
(usually aluminum oxide), and for exhausts of certain gelled liquid propellants, 
there is an impingement of these solid particles against the nozzle wall in the 
reversing curvature section between I and E in Fig. 3-14. While the gas can be 
turned by oblique waves to have less divergence, the particles (particularly the 
larger particles) have a tendency to move in straight lines and hit the walls at high 
velocity. The resulting abrasion and erosion of the nozzle wall can be severe, 
especially with the ablative and graphite materials that are commonly used. This 
abrasion by hot particles increases with turn-back angle. If the turn-back angle and 
thus also the inflection angle 0, are reduced, the erosion can become acceptable. 
Typical solid rocket motors flying today have values of inflection angles between 
20 and 26° and turn-back angles of 10 to 15°. In comparison, current liquid rocket 
engines without entrained particles have inflection angles between 27 and 50° and 
turn-back angles of between 15 and 30°. Therefore the performance enhancement 
caused by using a bell-shaped nozzle (high value of correction factor) is somewhat 
lower in solid rocket motors with solid particles in the exhaust. 

The ideal bell-shaped nozzle (minimum loss) is long, equivalent to a conical 
nozzle of perhaps 10 to 12°, as seen in Fig. 3-12. It has about the same length 
as a full-length aerospike nozzle. This is usually too long for reasonable vehicle 
mass ratios. 

Two-Step Nozzles. A two-position nozzle (with different expansion area 
ratios) can give a better performance than a conventional nozzle with a fixed 
single area ratio. This can be seen in Fig. 3-10; the lower area ratio nozzle 
(e = 6.0) performs best at low altitudes and the higher area ratio nozzle performs 
best at higher altitudes. If these two nozzles could somehow be mechanically 
combined, the resulting two-position nozzle would perform closer to an ideal 
nozzle which adjusts continuously to the optimum area ratio, as shown by the 
thin dashed curve. When integrated over the flight time, the extra performance 
can be noticeable and it has a pay-off for high-velocity missions, such as 
earth-orbit injection and deep space missions. Several concepts of a bell-shaped 
nozzle have evolved that achieve maximum performance at more than a single 
altitude. Figure 3-15 shows three concepts for a two-step nozzle, one that 
has an initial low area ratio A 2 jAt for operation at or near the earth’s surface 
and a larger second area ratio that improves performance at high altitudes. See 
Ref. 3-4. 

The extendible nozzle requires actuators, a power supply, mechanisms for mov- 
ing the extension into position during flight, and fastening and sealing devices. 
It has successfully flown in several solid rocket motor nozzles and in a few liq- 
uid engine applications, where it was deployed prior to ignition. Although only 



3.4. NOZZLE CONFIGURATIONS 83 



Extendible 
nozzle with 
two segments 
(shown in 
Fig. 8-17) 



Second nozzle exit segment 




chamber) after moving aft 
ready for operation 



Center line 



Droppable insert 
(mechanisms for holding, 
moving, or releasing the 
inserts are not shown) 




Chamber 




FIGURE 3-15. Simplified diagrams of three altitude-compensating two-step nozzle con- 
cepts. See Ref. 3-4. 



two steps are shown, there have been versions with three steps; one is shown 
in Fig. 12-3. The key concerns are a reliable rugged mechanism to move the 
extension into position, the hot gas seal between the nozzle sections, and the 
extra weight involved. Its principal merit is the short length of the nozzle while 
stored. This reduces the length of the vehicle and a considerable part of the inert 
vehicle mass. A further benefit is a significant performance gain, compared to a 
single nozzle of a length equal to the stored length. 

The droppable insert concept avoids the moving mechanism and gas seal but 
has a potential stagnation temperature problem at the joint. It requires a reliable 
release mechanism, and the ejected insert creates flying debris. To date it has 
only some actual test experience. See Refs. 3-4 and 3-12. 

The dual-bell nozzle concept uses two shortened bell nozzles combined into 
one with a bump or inflection point between them, as shown in Fig. 3-15. 
During ascent it functions first at the lower area ratio, with separation occurring 
at the bump. As altitude increases and the gas expands further, the flow attaches 
itself downstream of this point, with the flow filling the full nozzle exit section 
and operating with the higher area ratio at higher performance. There is a small 



84 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



performance penalty for a contour with a circular bump and concern about heat 
transfer. To date there has been little flight experience with this concept. 



Nozzles with Aerodynamic Boundaries 

These nozzles allow for nearly optimum nozzle expansion at all altitudes. The 
three nozzles shown on the right side of Fig. 3-12 offer full altitude compensa- 
tion and are discussed next. Refs. 3-4 and 3-8 give more information. Altitude 
compensation means that the nozzle allows nearly optimum performance at every 
altitude. Complete rocket engines with either a cut-off plug nozzle or an expan- 
sion/deflection nozzle have been developed and have been ground tested. Tests 
with both pressurized feed systems and turbopump feed systems have been suc- 
cessful. However, as of 2008, none of these rocket engines with aerodynamic 
nozzle boundaries have flown in a production vehicle. 

The plug nozzle or aerospike nozzle has an annular doughnut- shaped cham- 
ber with an annular nozzle slot. Annular nozzle throats may cause combustion 
problems. An alternate practical version has a number of individual small cham- 
bers (each with low area ratio short nozzles, a round throat, and a rectangular 
exit) arranged in a circle around a common plug or spike; see Fig. 8-12. The 
outside aerodynamic boundary of the gas flow in the divergent section of the 
nozzle is the interface between the hot gas and the ambient air; there is no outer 
wall as in a conical or bell-shaped nozzle. As the external or ambient pressure is 
reduced during the ascending flight, this gas boundary expands outward, causes 
a change in pressure distribution on the central spike, and allows an automatic 
and continuous altitude compensation. The aerospike contour with the minimum 
flow losses turns out to be very long, similar in length to an optimum bell nozzle 
as shown in Figs. 3-12 and 3-13. The mass flow per unit exit area is relatively 
uniform over the cross section and the divergence losses are minimal. 

If the central plug is cut off or truncated and the wall contour is slightly 
altered, then the nozzle will be very short, as shown in Figs. 3-12, 3-13, and 
3-14. It will have some internal supersonic waves and will show a small but 
real loss in thrust compared to a nozzle with a full length central spike. There is 
a small performance loss due to shock waves generated by the impingement of 
gases from the adjacent small chambers. The pressure distribution and the heat 
transfer intensity vary on the inner contoured spike wall surface and vary with 
altitude. Figure 8-12 shows a typical pressure distribution over the contoured 
spike surface at high and low altitudes. 

The pressure in the recirculating trapped gas of the subsonic region below the 
bottom plate also exerts a thrust force. The losses caused by the cut-off spike 
can be largely offset by injecting a small amount of the gas flow (about 1% of 
total flow) through this base plate into the recirculating region, thus enhancing 
the back pressure on the base plate. The advantages of the truncated aerospike 
are short length (which helps to reduce the length and mass of the flight vehicle), 
full altitude compensation, no flow separation from the wall at lower altitudes, 
and ease of vehicle/engine integration for certain vehicle configurations. 



3.5. REAL NOZZLES 85 



The linear aerospike nozzle is a variation of the round axisymmetric aerospike 
nozzle. Basically, it is an unrolled version of the circular configuration. It is 
explained further in Section 8.2. 

In the expansion deflection nozzle (Fig. 3-12) the flow from the chamber 
is directed radially outward away from the nozzle axis. The flow is turned on 
a curved contour outer diverging nozzle wall. The nozzle has been shortened 
and has some internal oblique shock wave losses. The hot gas flow leaving the 
chamber expands around a central plug. The aerodynamic interface between the 
ambient air and gas flow forms an inner boundary of the gas flow in the diverging 
nozzle section. As the ambient pressure is reduced, the hot gas flow fills more 
and more of the nozzle diverging section. Altitude compensation is achieved by 
this change in flow boundary and by changes in the pressure distribution on the 
outer walls. Actual ground tests with both liquid engines and solid motors have 
not yet led to routine flight tests. 

Multiple Nozzles. If a single large nozzle is replaced by a cluster of smaller 
nozzles on a solid motor (with the same total thrust), then it is possible to reduce 
the nozzle length. Similarly, if a single large thrust chamber of a liquid engine 
is replaced by several smaller thrust chambers, the nozzle length will be shorter, 
reducing the vehicle length and thus the vehicle structure and inert mass. Reaction 
Motors, Inc. (RMI), in their first aircraft-rocket engine, and the Soviets pioneered 
a set of four thrust chambers, each with 25% of the total thrust, assembled next to 
each other and fed from the same liquid propellant feed system. This quadruple 
thrust chamber arrangement has been used effectively on many large Russian 
space launch vehicles and missiles. As seen in Fig. 3-13, the nozzle length of 
this cluster is about 30% shorter than a single nozzle of an equivalent larger thrust 
chamber. The vehicle diameter at the cluster nozzle exit is somewhat larger, the 
vehicle drag is higher, and there is additional engine complexity and often more 
engine mass. 



3.5. REAL NOZZLES 

The assumptions and simplifications listed in Section 3.1 are only approxima- 
tions that allow relatively simple algorithms and simple mathematical solutions 
to the analysis of real rocket nozzle phenomena. For most of these assumptions 
it is possible either (1) to use an empirical correction factor (based on experi- 
mental data) or (2) to develop or use a more accurate algorithm, which involves 
more detailed understanding and simulation of energy losses, the physical or 
chemical phenomena, and also often a more complex theoretical analysis and 
mathematical treatment. Some of these approaches are mentioned briefly in this 
section. 

Compared to an ideal nozzle, the real nozzle has energy losses and energy that 
is unavailable for conversion into kinetic energy of the exhaust gas. The principal 
losses are listed below and several of these are discussed in more detail. 



86 



NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



1. The divergence of the flow in the nozzle exit sections causes a loss, which 
varies as a function of the cosine of the divergence angle as shown by 
Eq. 3-34 and Table 3-3 for conical nozzles. The losses can be reduced 
for bell-shaped nozzle contours. 

2. Small chamber or port area cross sections relative to the throat area or low 
nozzle contraction ratios A\/A, cause pressure losses in the chamber and 
reduce the thrust and exhaust velocity slightly. See Table 3-2. 

3. Lower flow velocity in the boundary layer or wall friction can reduce the 
effective exhaust velocity by 0.5 to 1.5%. 

4. Solid particles or liquid droplets in the gas can cause losses perhaps up to 
5%, as described below. 

5. Unsteady combustion and oscillating flow can account for a small loss. 

6. Chemical reactions in nozzle flow change gas composition and gas proper- 
ties and gas temperatures, giving typically a 0.5% loss. See Chapter 5. 

7. There is lower pressure and lower performance during transient operation, 
for example, during start, stop, or pulsing. 

8. For uncooled nozzle materials, such as fiber-reinforced plastics or carbon, 
the gradual erosion of the throat region increases the throat diameter by 
perhaps 1 to 6% during operation. In turn this will reduce the chamber 
pressure and thrust by about 1 to 6%. This enlargement of the throat area 
causes a drop in chamber compression and a slight reduction in specific 
impulse usually less than 0.7%. 

9. Nonuniform gas composition can reduce performance (due to incomplete 
mixing, turbulence, or incomplete combustion). 

10. Using real gas properties can at times change the gas composition, the 
values of k and SOI, and this can cause a small loss in performance, say 0.2 
to 0.7%. 

11. Operation at nonoptimum nozzle expansion area ratio reduces thrust and 
specific impulse. There is no loss if the vehicle only flies at the altitude 
for optimum nozzle expansion (j >2 — Pi,)- If it flies with a fixed nozzle area 
ratio at higher or lower altitudes, then there is a loss (during a portion of the 
flight) by up to 15% in thrust compared to a nozzle with altitude compen- 
sation, as can be seen in Figs. 3-7 and 3-8. It also reduces performance 
by 1 to 5%. 



Boundary Layer 

Real nozzles have a viscous boundary layer next to the nozzle walls, where the 
gas velocities are much lower than the free-stream velocities in the inviscid flow 
regions. An enlarged schematic view of a boundary layer is shown in Fig. 3-16. 
Immediately next to the wall the flow velocity is zero and then the boundary layer 
can be considered as being built up of successive annular-shaped thin layers of 
increasing velocity until the free-stream velocity is reached. The low-velocity 



3.5. REAL NOZZLES 87 




^ . 

Boundary. 

layer _ 
thickness" 



Wall thickness 

1 Vw = 0 









Velocity profile 



Nozzle wall 



1 


















3 















Temperature profile 



FIGURE 3-16. Flow conditions at a nozzle exit lip at high altitude, showing streamlines, 
boundary layer, velocity, and temperature profiles. 



flow close to the wall is laminar and subsonic, but in the higher-velocity regions 
of the boundary layer the flow is supersonic and can become turbulent. The 
local temperature in part of the boundary layer can be substantially higher than 
the free-stream temperature because of the eonversion of kinetie energy into 
thermal energy as the local velocity is slowed down and as heat is created by 
viscous friction. The layer right next to the wall will be cooler because of heat 
transfer to the wall. The gaseous boundary layer has a profound effect on the 
overall heat transfer to nozzle and chamber walls. It also has an effect on the 
rocket performance, particularly in applications with relatively long nozzles with 
high nozzle area ratios, where a relatively high proportion of the total mass 
flow (2 to 25%) can be in the lower- velocity region of the boundary layer. 
The high gradients in pressure, temperature, or density and the changes in local 
velocity (direction and magnitude) influence the boundary layer. Scaling laws for 
boundary layer phenomena have not been reliable. 

Theoretical approaches to boundary layer performance effects can be found in 
Chapters 26 to 28 of Ref. 3-1 and in Ref. 3-17. A truly satisfactory theoretical 
analysis of boundary layers in rocket nozzles has not yet been developed. For- 
tunately, the overall effect of boundary layers on rocket performance has been 
small. For most rocket nozzles the loss seldom exceeds 1% of specific impulse. 



88 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Multiphase Flow 

In some propulsion systems the gaseous working fluid contains small liquid 
droplets and/or solid particles that must be accelerated by the gas. They give up 
heat to the gas during the expansion in a nozzle. This, for example, occurs with 
solid propellants (see Chapter 13) or some gelled liquid propellants (Section 7.5), 
which contain aluminum powder that forms small oxide particles in the exhaust. 
It can also occur with iron oxide catalysts, or propellants containing beryllium, 
boron, or zirconium. 

In general, if the particles are very small (typically with diameters of 0.005 
mm or less), they will have almost the same velocity as the gas and will be in 
thermal equilibrium with the nozzle gas flow. Thus, as the gases give up kinetic 
energy to accelerate the particles, they gain thermal energy from the particles. 
As the particle diameters become larger, the mass (and thus the inertia) of the 
particle increases as the cube of its diameter; however, the drag force increases 
only as the square of the diameter. Larger particles therefore do not move as 
fast as the gas and do not give heat to the gas as readily as do smaller particles. 
The larger particles have a lower momentum than an equivalent mass of smaller 
particles and they reach the nozzle exit at a higher temperature than the smaller 
particles, thus giving up less thermal energy. 

It is possible to derive a simple theoretical approach for correcting the perfor- 
mance (/j,c, or c*) as shown below and as given in Refs. 3-13 and 3-14. It is 
usually based on the following assumptions: specific heats of the gases and the 
particles are constant throughout the nozzle flow, the particles are small enough 
to move at essentially the same velocity as the gas and are in thermal equilibrium 
with the gas, and that particles do not exchange mass with the gas (no vapor- 
ization or condensation). Expansion and acceleration occur only in the gas, and 
the volume occupied by the particles is negligibly small compared to the gas 
volume. If the amount of particles is small, the energy needed to accelerate the 
particles can also be neglected. There are no chemical reactions. 

The enthalpy h, the specific volume V, the specifie heat notes k, and the gas 
constant R can be expressed as functions of the particle fraction p, which is 
the mass of particles (liquid and/or solid) divided by the total mass. Using the 
subscripts g and 5 to refer to the gas or solid state, the following relationships 
then apply: 



= {1 - P)(Cp)gT + pc,T 


(3-35) 


II 

1 

3 


(3-36) 


= RJ/V, 


(3-37) 


= (1 - P)Rs 


(3-38) 


(1 - P)Cp -b PCs 




(1 — P)c„ + PCg 


w 



3.5. REAL NOZZLES 89 



These relations are then used in the formulas for simple one-dimensional nozzle 
flow, such as Eq. 2-16, 3-15, or 3-32. The values of specific impulse or char- 
acteristic velocity will decrease as p, the percent of particles, is increased. For 
very small particles (less than 0.01 mm in diameter) and small values of /I (less 
than 6%) the loss in specific impulse is often less than 2%. For larger particles 
(over 0.015 mm diameter) and larger values of ^ this theory is not helpful and 
the specific impulse can be 10 to 20% less than the 4 value without flow lag. 
The actual particle sizes and distribution depend on the specific propellant, the 
combustion, the particular particle material, and the specific rocket propulsion 
system, and usually have to be measured (see Chapters 13 and 20). Thus adding 
a metal, such as aluminum, to a solid propellant will increase the performance 
only if the additional heat release can increase the combustion temperature T\ 
sufficiently so that it more than offsets the decrease caused by the nonexpanding 
mass of the particles in the exhaust. 

With very high area ratio nozzles and a low nozzle exit pressure (high altitude 
or space vacuum) it is possible to condense some of the propellant ingredients 
that are normally gases. As the temperature drops sharply in the nozzle, it is 
possible to condense gaseous species such as H2O, CO2, or NH3 and form liquid 
droplets. This causes a decrease in the gas flow per unit area and the transfer 
of the latent heat of vaporization to the remaining gas. The overall effect on 
performance is small if the droplet size is small and the percent of condensed 
gas mass is moderate. It is also possible to form a solid phase and precipitate 
fine particles of snow (H2O) or frozen fog of other species. 



Other Phenomena and Losses 

The combustion process is really not steady. Low- and high-frequency oscillations 
in chamber pressure of up to perhaps 5% of rated value are usually considered 
as smooth-burning and relatively steady flow. Gas properties {k,DJl,Cp) and flow 
properties (v, V,T,p, etc.) will also oscillate with time and will not necessarily 
be uniform across the flow channel. These properties are therefore only “aver- 
age” values, but it is not always clear what kind of an average they are. The 
energy loss due to nonuniform unsteady burning is difficult to assess theoret- 
ically. For smooth-burning rocket systems they are negligibly small, but they 
become significant for larger-amplitude oscillations. 

The composition of the gas changes somewhat in the nozzle, chemical reac- 
tions occur in the flowing gas, and the assumption of a uniform or “frozen” 
equilibrium gas composition is not fully valid. A more sophisticated analysis for 
determining performance with changing composition and changing gas properties 
is described in Chapter 5. The thermal energy that is carried out of the nozzle 
(mCpT 2 ) is unavailable for conversion to useful propulsive (kinetic) energy, as is 
shown in Fig. 2-3. The only way to decrease this loss is to reduce the nozzle 
exit temperature T 2 (larger nozzle area ratio), but even then it is a large loss. 

When the operating durations are short (as, e.g., with antitank rockets or 
pulsed attitude control rockets which start and stop repeatedly), the start and 



90 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Stop transients are a significant portion of the total operating time. During the 
transient periods of start and stop the average thrust, chamber pressure, or specific 
impulse will be lower in value than those same parameters at steady full operating 
conditions. This can be analyzed in a step-by-step process. For example, during 
startup the amount of propellant reacting in the chamber has to equal the flow 
of gas through the nozzle plus the amount of gas needed to fill the chamber to 
a higher pressure; alternatively, an empirical curve of chamber pressure versus 
time can be used as the basis of such a calculation. The transition time is very 
short in small, low-thmst propulsion systems, perhaps a few milliseconds, but it 
can be longer (several seconds) for large propulsion systems. 



Performance Correction Factors 

Additional to the theoretical cone- and bell-shaped nozzle correction factors dis- 
cussed in Section 3.4., we discuss here a set of empirical correction factors. These 
factors represent a variety of nonideal phenomena (such as friction, imperfect 
mixing and combustion, heat transfer, chemical nonequilibrium, and two- and 
three-dimensional effects) which are unavoidably present; refer to Fig. 2-3 and 
Ref. 3-4. Correction factors are defined for convenience in rocket analysis and 
may differ from the more conventional efficiencies — there are no “universal per- 
formance correction factors.” For specific propulsion systems, where accurately 
measured data are available, they allow for simple predictions of the actual perfor- 
mance. For example, a velocity correction factor of 0.942 means that the velocity 
or the actual specific impulse is about 94% of theoretical (the commonly accepted 
value might be closer to 0.92). 

Corrections factors are used by engineers to predetermine performance ahead 
of testing, and for preliminary designs, informal proposals, and in health moni- 
toring systems. For all these, a set of accepted or nominal values is needed to 
estimate performance together with some useful formula-recipes. For the most 
accurate calculations, the industry relies extensively on sophisticated computer 
programs which are largely proprietary. 

Correction factors are ratios of the measured or actual (subscript “a”) to 
the formulated or ideal (subscript “i”) values. In the testing of propulsion sys- 
tems, the combustion chamber pressure, the propellant mass flow rates, the thrust 
force, and the throat and exit areas are typically measured. These measurements 
yield two direct ratios, namely the thrust correction factor {t^f = Fa/Fj) and the 
discharge correction factor = tha/tkj), and as well the product [{pi)a(At)a] 
which has the units of force. This product enters into the formulation of two other 
correction factors discussed below. The nozzle area ratio [Ca = (A 2 )a/(A,)a] in 
real nozzles may be found from their measured dimensions and this ratio will 
differ from the ideal calculated value as will be shown in Example 3-7. 

The thrust correction factor (f^ ) is found from the ratio of thrust measurements 
with the corresponding ideal values of Eq. 3-29. The discharge correction factor 
{t^d) can be determined from the ratio of mass-flow-rate measurements with the 
corresponding theoretical values of Eq. 3-24. Unlike incompressible flows, in 



3.5. REAL NOZZLES 91 



rocket propulsion systems the value of is always somewhat larger than 1.0 
(up to 1.15). The actual flow rate may he larger than the ideal flow rate for the 
following reasons: 

1 . Incomplete combustion (a lower combustion temperature) which results in 
increases of the exhaust gas densities. 

2. Cooling at the walls which reduces the boundary layer temperature and 
thus the average gas temperature, especially in small thrust chambers. 

3. Changes in the specific heat ratio and molecular mass in an actual nozzle 
which affect the flow rate and thus the discharge correction factor (see 
Eq. 3-24). 

The c* efficiency or fc* correction factor represents the combined effective- 
ness of the combustion chamber and the injector design. It can be determined 
from the ratio of measured values of [ipi)a{At)ay ,ma (from Eq. 2-18) with the 
corresponding ideal value of the right-hand side of Eq. 3-32. In well-designed 
combustion chambers, the value of fc* correction factor is over 95%. The 
correction factor , also known as the Cp efficiency, represents the effectiveness 
of the nozzle design at the operating conditions. It can be determined using mea- 
sured values (from Eq. 3-31) of Fa/[ip\)a{^i)a^ with the corresponding ideal 
value of Eq. 3-30. In well-designed nozzles, the value of the fcp correction 
factor is above 90%. 

An effective exhaust velocity correction factor [fy = (Falrna)lci\ may now be 
introduced using that velocity’s definition given in Eq. 2-17 (or from Eq. 3-32, 
c = c*Cf) as 



Kv = KpfKd = Kc*Kcp (3-40) 

This further suggests a correction factors relation, a form equivalent to Eq. 3-33, 
written as fp = t,dKc*Kcp which must hold only within experimental uncertainties. 

Additional useful relations may now be written. The actual specific impulse, 
from Eq. 2-5, may now be calculated from 



{ls)a = {ls)i{K.) (3-41) 

The thermodynamic nozzle efficiency which is defined as the ratio of ideal to 
the actual enthalpy changes (see Eqs. 3- 15a and 3-16) under a given pressure 
ratio becomes 



r]„ = (Ah)a/iAh)i = \{vlf)al\{vif)i k. (f„)2 (3-42) 

The approximation sign becomes an equality when p 2 = p^. This nozzle effi- 
ciency will be always less than one and represents losses inside the nozzle 
(Ref. 3-17). Small nozzles being used in micropropulsion can be quite inef- 
ficient because of their relatively larger frictional effects which are in proportion 



92 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



to the surface-to-volume ratio (some being rectangular thus having additional 
sharp-corner losses). 

Under &tnctly frozen-flow assumptions, the right-hand side of Eq. 3-32 leads 
to a useful relation between the ideal and actual stagnation temperatures [where 

(To)i ^Ti]: 



(To)a/iTofl « (3-43) 

With conditions at the inlet of the nozzle readily calculable, one task is to 
determine the actual throat area required to pass a specified mass flow rate of 
gaseous propellant and also to determine the area and the flow properties at 
the nozzle exit plane. Across the nozzle heat losses modify the local stagnation 
temperatures and, together with friction, the stagnation pressures (which begin as 
(Pq)i = Pi and Ti in the combustion chamber). Because ideal formulations are 
based on the local stagnation values at any given cross section, the challenge is 
to work with the appropriate flow assumptions as shown below. 

When the applicable correction factors values are available, the overall ratio 
of the product of the stagnation pressure with the throat area may be inferred 
using 



[{At)a(P\)a]/[{At)i(pi)i] = or ^d^c* (3-44) 

In order to arrive at the actual nozzle throat area from such available parameters 
additional information is needed. The key question is how to relate stagnation 
pressure ratios to their corresponding temperature ratios across the nozzle. When 
known, a poly tropic index “n” providing a relation between gas properties is 
useful (Ref. 3-17); otherwise, we may apply the isentropic relation given as 
Eq. 3-7 because nozzle efficiencies are typically high (around 90%). Real noz- 
zle flows are not isentropic but the net entropy change along the nozzle may be 
insignificant, though noticeable decreases of the stagnation pressure and temper- 
ature do exist as Example 3-7 will detail next. When dealing with real nozzles 
with given expansion ratios, increases of stagnation pressure over (pi), would 
be necessary to achieve a desired performance. 

Example 3-7. Design a rocket nozzle to conform to the following conditions; 

Chamber pressure 20.4 atm = 2.067 MPa 

Atmospheric pressure 1.0 atm 

Mean molecular mass of gases 21.87 kg/kg-mol 

Specific heat ratio 1.23 

Ideal specific impulse 230 sec (at operating conditions) 

Desired thrust 1300 N 

Chamber temperature 2861 K 



3.5. REAL NOZZLES 93 



Determine the actual exhaust velocity, the actual specific impulse, and the nozzle throat 
and exit areas. Also calculate the discharge correction factor and nozzle efficiency implied 
by the following correction factors which are applicable to this problem: = 0.96, fc* = 

0.98, and = 0.97. 

SOLUTION. The theoretical thrust coefficient can be calculated from Eq. 3-30. For 
optimum conditions, p 2 = P 3 - By substituting k = 1.23 and pi/p 2 = 20.4, the thrust coef- 
ficient is (Cf ); = 1.405 and (A,)/ = 4.48 cm^. These calculated values may be checked 
by interpolation between the values shown in Fig. 3-6. 

An actual effective exhaust velocity is not measured, but for this problem it may be 
calculated from 



Ca = (Ci)(iv) = (230 X 9.81)(0.98 x 0.97) = 2145 m/sec 

The actual specific impulse is (Da = 2145/9.81 = 219 sec and the discharge correction 
factor from Eq. 3-40 

(d = Kf/Kc*Kc, = 0.96/(0.98 x 0.97) = 1.01 
Because for this problem, V 2 = c, the nozzle efficiency is found from r/„ = 

(f„)2 = 90%. 

An approximate exit stagnation temperature may be found using Eq. 3-43, 
(7b)a ^ = 2748 K. Now, an estimate of the stagnation pressure at the throat 

is needed but correction factors only apply to the overall nozzle. Because the nozzle 
efficiency is only 10% less than ideal, a sufficient approximation is to assign half of the 
stagnation temperature drop to the region upstream of the throat, or (To)f ^ 2804 K. 
Thus, 

(po), « pi[2804/2861](*'"'-^) = 1.86 MPa 
The throat area may now be computed from Eq. 3-44: 

(At)a/(At)i = ii;F/ilCF)(Pl/(Po)t) = 1.10 

As may have been expected, the actual throat area is about 10% larger than the ideal for 
the same flow conditions. The actual exit area may be obtained going back to the mass 
flow rate: 

rh = t,pF I Ca = 0.582 kg/sec = P 2 MC/RT 2 

Now, p 2 = P 3 is given and T 2 is found from 

T 2 « 2748 X (0.1013/1.67)<°-22®''‘-22®) = 1631 K 

where (po)a ^ 1-67 MPa is an actual stagnation pressure applicable to the exit of the 
real nozzle. Finally, the exit area and area ratio are found to be. (A 2 )a 16.61 cm^ and 

Ca ^ 3.37, which may be compared to the value from Fig. 3-4. 



94 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



Four Performance Parameters 

When using values or quoting of thrust, specific impulse, propellant flow, and 
other performance parameters, one must be careful to specify or qualify the 
conditions under which a specific number is presented. There are at least four 
sets of performance parameters and they are different in concept and value, 
even when referring to the same rocket propulsion system. Each performance 
parameter, such as F, 7^, c, V 2 , and/or hi, should be accompanied by a clear 
definition of the conditions under which it applies. Not all the items below apply 
to every one of the parameters. 

(a) Chamber pressure; also, for slender chambers, the location where this pres- 
sure prevails or is measured (e.g., at nozzle entrance). 

(b) Ambient pressure or altitude or space (vaeuum). 

(c) Nozzle expansion area ratio and whether this is an optimum. 

(d) Nozzle shape and exit angle (see Table 3-3 and Fig. 3-12). 

(e) Propellants, their composition or mixture ratio. 

(f) Key assumptions and corrections made in the calculations of the theoretical 
performance: for example, was frozen or shifting equilibrium used in the 
analysis? (This is described in Chapter 5.) 

(g) Initial ambient temperature of propellants, prior to start. 

1. Theoretical performance values are defined in Chapters 2, 3, and 5 and 
generally apply to ideal rockets. The conditions for an ideal nozzle were given 
in Section 3.1. The theoretical performance of an ideal nozzle per se is not 
used, but one that includes theoretical corrections is used. Analyses of one or 
more losses can be included, and these will yield a lower theoretical perfor- 
mance and also an assessment in the correction factors described in the previous 
section. Most are two dimensional and correct for the chemical reactions in the 
nozzle using real gas properties, and most correct for divergence. Many also 
correct for one or more of the other losses mentioned above. For example, pro- 
grams for solid propellant motor nozzles can include losses for throat erosion 
and multiphase flow; for liquid propellant engines it may include two or more 
concentric zones, each at different mixture ratios and thus with different gas 
properties. Nozzle wall contour analysis with expansion and compression waves 
may use a finite element analysis and/or a method of characteristics approach. 
Some of the more sophisticated programs include analyses of viscous boundary 
layer effects and heat transfer to the walls. Typically these computer simulation 
programs are based on computer fluid dynamics finite element analyses and on 
the basic Navier-Stokes relationships. Most organizations doing nozzle design 
have their own computer programs, often different programs for different nozzle 
designs, different thrust levels, or operating durations. Many also have simpler. 



3.5. REAL NOZZLES 95 



one-dimensional computer programs which may include one or more of the 
above corrections; they are used frequently for preliminary estimates or informal 
proposals. 

2. Delivered, that is, actually measured, performance values are obtained from 
static tests or flight tests of full-scale propulsion systems. Again, the conditions 
should be explained (e.g., define pi,A 2 /At,p^, etc.) and the measured values 
should be corrected for instrument deviations, errors, or calibration constants. 
Flight test data need to be corrected for aerodynamic effects, such as drag. Often 
empirical coefficients, such as the thrust correction factor, the velocity correction 
factor, and the mass discharge flow correction factors are used to convert the 
theoretical values of item 1 above to approximate actual values and this is often 
satisfactory for preliminary estimates. Sometimes subscale thrust chambers have 
been used in the development of new thrust chambers and then scale factors are 
used to correct the measured data to full-scale values. 

3. Performance values at standard conditions are corrected values of items 
1 and 2 above. These standard conditions are generally rigidly specified by the 
customer and by commonly accepted industrial practice. Usually they refer to 
conditions that allow ready evaluation or comparison with reference values and 
often they refer to conditions that can be easily measured and/or corrected. For 
example, to allow a good comparison of specific impulse for several propellants 
or rocket propulsion systems, the values are often corrected to the following 
standard conditions (see Example 3-6): 

(a) Pi = 1000 psia or 6.894 x 10^ Pa. 

(b) p 2 = p 3 = 14.69 psia (sea level) or 1.0132 x 10^ Pa or 0.10132 Mpa. 

(c) Nozzle exit area ratio is optimum, p 2 = P 3 - 

(d) Nozzle divergence half angle a = 15° for conical nozzles, or some agreed- 
upon value. 

(e) Specific propellant, its design mixture ratio, and/or propellant composition. 

(f) Propellant initial ambient temperature: 21°C (sometimes 20 or 25°C) or boil- 
ing temperature, if cryogenic. 

A rocket propulsion system is generally designed, built, tested, and delivered 
in accordance with some predetermined requirements or specifications, usually in 
formal documents often called the rocket engine or rocket motor specifications . 
They may define the performance as shown above and they also define many 
other requirements. More discussion of these specifications is given as a part of 
the selection process for propulsion systems in Chapter 19. 

4. Rocket manufacturers are often required by their customers to deliver 
rocket propulsion systems with a guaranteed minimum performance , such as 
minimum F or minimum f and/or both. The determination of this value can 
be based on a nominal value (items 1, 2, or 3 above) diminished by all likely 
losses, including changes in chamber pressure due to variation of pressure drops 
in injector or pipelines, a loss due to nozzle surface roughness, propellant initial 



96 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



ambient temperatures, ambient pressure variations, and manufacturing variations 
from rocket to rocket (e.g., in grain volume, nozzle dimensions, or pump impeller 
diameters, etc.)- This minimum value can be determined by a probabilistic 
evaluation of these losses and is then usually validated by actual full-scale static 
and flights tests. 



3.6. NOZZLE ALIGNMENT 

When the thrust line or direction does not intersect the center of mass of a 
flying vehicle, a turning moment will tend to rotate a vehicle in flight. Turning 
moments are desirable and necessary for the controlled turning or attitude control 
of a vehicle as is routinely done by means of the deflection of the thrust vector, 
aerodynamic fins, or by separate attitude control rocket engines. However, this 
turning is undesirable when its magnitude or direction is not known; this happens 
when a fixed nozzle of a major propulsion system has its thrust axis misaligned. 
A large high-thrust booster rocket system, even if misaligned by a very small 
angle (less than 0.50°), can cause major upsetting turning moments for the firing 
duration. If not corrected or compensated, such a small misalignment can cause 
the flight vehicle to tumble and/or deviate from the intended flight path. For 
this moment not to exceed the vehicle’s compensating attitude control capability, 
it is necessary to align the nozzle axis of all propulsion systems with fixed 
(nongimbal) nozzles very accurately. Normally, the geometric axis of the nozzle 
diverging exit surface geometry is taken to be the thrust axis. Special alignment 
fixtures are usually needed to orient the nozzle axis to be within less than ±0.25° 
of the intended line to the vehicle’s center of gravity and to position the center 
of a large nozzle throat to be on the vehicle centerline, say within 1 or 2 mm. 
See Ref. 3-16. 

There are other types of misalignments: (1) irregularities in the nozzle geom- 
etry (out of round, protuberances, or unsymmetrical roughness in the surface); 
(2) transient flow misalignments during start or stop; (3) uneven deflection of the 
propulsion system or vehicle structure under load; and (4) irregularities in the 
gas flow (faulty injector, uneven burning rate in solid propellants). For simple 
unguided rocket vehicles it has been customary to rotate or spin the vehicle to 
prevent the misalignment from being in one direction only or to even out the 
misalignment during powered flight. 

In the cramped volume of spacecraft or upper stage launch vehicles, it is 
sometimes not possible to accommodate the full length of a large-area-ratio nozzle 
within the available vehicle envelope. In this case the nozzles are cut off at an 
angle at the vehicle surface, which allows a more compact installation. Figure 
3-17 shows a diagram of two (out of four) roll control thrusters whose nozzle 
exit conforms to the vehicle contour. The thrust direction of a scarfed nozzle is 
no longer on the nozzle axis centerline, as it is with fully symmetrical nozzles, 
and the nozzle exit flow will not be axisymmetric. Reference 3-16 shows how 
to estimate the performance and thrust direction of scarfed nozzles. 



PROBLEMS 97 



Geometric centerline of nozzle 




FIGURE 3-17. Simplified partial section of a flight vehicle showing two attitude control 
thrusters with scarfed nozzles to fit a cylindrical vehicle envelope. 



PROBLEMS 

1. Certain experimental results indicate that the propellant gases of a liquid oxygen- 
gasoline reaction have a mean molecular mass of 23.2 kg/kg-mol and a specific heat 
ratio of 1.22. Compute the specific heat at constant pressure and at constant volume, 
assuming a perfect gas. 

2. The actual conditions for an optimum expansion nozzle operating at sea level are given 
below. Calculate V 2 , Tz, and Cf- The mass flow m = 3.7 kg/sec; pi = 2.1 MPa; T\ = 
2585 K; 9J1 = 18.0 kg/kg-mol; and k = 1.30. 

3. A certain nozzle expands a gas under isentropic conditions. Its chamber or nozzle 
entry velocity equals 90 m/sec, its final velocity 1500 m/sec. What is the change in 
enthalpy of the gas? What percentage of error is introduced if the initial velocity is 
neglected? 

4. Nitrogen at 500°C (k = 1.38, molecular mass is 28.00) flows at a Mach number of 
2.73. What are its actual and its acoustic velocity? 



98 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



5. The following data are given for an optimum rocket: 



Average molecular mass 



24 kg/kg-mol 
2.533 Mpa 
0.090 Mpa 
2900 K 



Chamber pressure 
External pressure 



Chamber temperature 
Throat area 
Specific heat ratio 



0.00050 m- 
1.30 



,2 



Determine (a) throat velocity; (b) specific volume at throat; (c) propellant flow and 
specific impulse; (d) thrust; (e) Mach number at throat. 

6. Determine the ideal thrust coefficient for Problem 5 by two methods. 

7. A certain ideal rocket with a nozzle area ratio of 2.3 and a throat area of 5 in.^ delivers 
gases at = 1.30 and R = 66 ft-lbf/lbm-°R at a chamber pressure of 300 psia and 
a constant chamber temperature of 5300°R against a back atmospheric pressure of 
10 psia. By means of an appropriate valve arrangement, it is possible to throttle the 
propellant flow fo the thrust chamber. Calculate and plot against pressure the following 
quantities for 300, 200, and 100 psia chamber pressure: (a) pressure ratio between 
chamber and atmosphere; (b) effective exhaust velocity for area ratio involved; (c) 
ideal exhaust velocity for optimum and actual area ratio; (d) propellant flow; (e) 
thrust; (f) specific impulse; (g) exit pressure; (h) exit temperature. 

8. For an ideal rocket with a characteristic velocity c* = 1500 m/sec, a nozzle throat 
diameter of 20 cm, a thrust coefficient of 1.38, and a mass flow rale of 40 kg/sec, 
compute the chamber pressure, the thrust, and the specific impulse. 

9. For Ihe rockel unit given in Example 3-2 compute the exhaust velocity if the nozzle 
is cut off and the exit area is arbitrarily decreased by 50%. Estimate the losses in 
kinetic energy and thrust and express them as a percentage of the original kinetic 
energy and the original thrust. 

10 . What is the maximum velocity if the nozzle in Example 3-2 was designed to expand 
into a vacuum? If the expansion area ratio was 2000? 

11 . Construction of a variable-area conventional axisymmetric nozzle has often been 
considered to make the operation of a rocket thrust chamber take place at the optimum 
expansion ratio at any altitude. Because of the enormous design difficulties of such 
a mechanical device, it has never been successfully realized. Assuming that such a 
mechanism can eventually be constructed, what would have to be the variation of the 
area ratio with altitude (plot up to 50 km) if such a rocket had a chamber pressure 
of 20 atm? Assume that k = 1 .20. 

12 . Design a supersonic nozzle to operate at 10 km altitude with an area ratio of 8.0. 
For the hot gas take To = 3000 K,R = 378 J/kg-K, and k = 1.3. Determine the exit 
Mach number, exit velocity, and exit temperature, as well as the chamber pressure. 
If this chamber pressure is doubled, what happens to the thrust and the exit velocity? 
Assume no change in gas properties. How close to optimum nozzle expansion is this 
nozzle? 



PROBLEMS 99 



13. The German World War II A-4 propulsion system had a sea-level thrust of 25,400 kg 
and a chamber pressure of 1.5 Mpa. If the exit pressure is 0.084 MPa and the exit 
diameter 740 mm, what is the thrust at 25,000 m? 

14. Derive Eq. 3-34. {Hint: Assume that all the mass flow originates at the apex of 
the cone.) Calculate the nozzle angle correction factor for a conical nozzle whose 
divergence half angle is 13°. 

15. For Example 3-2, determine (a) the actual thrust; (b) the actual exhaust velocity; (c) 
the actual specific impulse; (d) the velocity correction factor. Assume that the thrust 
correction factor is 0.985 and the discharge correction factor is 1.050. 

16. An ideal rocket has the following characteristics: 



Determine the critical pressure ratio, the gas velocity at the throat, the expansion area 
ratio, and the theoretical nozzle exit velocity. 

Answers: 0.5645; 3470 ft/sec; 14.8; and 8570 ft/sec. 

17. For an ideal rocket with a characteristic velocity c* of 1220 m/sec, a mass flow rate 
of 73.0 kg/sec, a thrust coefficient of 1.50, and a nozzle throat area of 0.0248 m^, 
compute the effective exhaust velocity, the thrust, the chamber pressure, and the 
specific impulse. 

Answers: 1830 m/sec; 133,560 N; 3.590 x 10® N/m^; 186.7 sec. 

18. Derive Eqs. 3-24 and 3-25. 

19. An upper stage of a launch vehicle propulsion unit fails to meet expectations during 
sea-level testing. This unit consists of a chamber at 4.052 MPa feeding hot propellant 
to a supersonic nozzle of area ratio e = 20. The local atmospheric pressure at the 
design condition is 20 kPa. The propellant has a. k = 1.2 and the throat diameter of 
the nozzle is 9 cm. 

(a) Calculate the ideal thrust at the design condition. 

(b) Calculate the ideal thrust at the sea-level condition. 

(c) State the most likely source of the observed nonideal behavior. 

Answers: (a) 44.4 kN, (b) 34.1 kN, (c) separation in the nozzle 

20. Assuming ideal flow within the propulsion unit: 

(a) State all necessary conditions (realistic or not) for 



Chamber pressure 
Nozzle exit pressure 
Specifle heat ratio 
Average molecular mass 
Chamber temperature 



1.20 

21.0 Ibm/lb-mol 
4200°F 



27.2 atm 
3 psia 



c* = c = l >2 



(b) Do the above conditions result in an optimum thrust for a given pi/ps? 



100 NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



(c) For a launch vehicle designed to operate at some intermediate earth altitude, 
sketch (in absolute or relative values) how c*, c, and V 2 would vary with altitude. 

21 . A rocket nozzle has been designed with A, = 19.2 in.^ and A 2 = 267 in.^ to operate 
optimally at p 3 = 4 psia and produce 18,100 Ibf of ideal thrust with a chamber pres- 
sure of 570 psia. It will be attached to a previously built thrust chamber that operates 
at T\ = 6000°R with k = 1.25 and R = 68.75 ft-lbf/lbm°R, with a c* efficiency of 
95%. But test measurements on this thrust system, at the stated pressure conditions, 
yield a thrust of only 16,300 Ibf when the measured flow rate is 2.02 Ibm/sec. Find the 
applicable correction factors and the actual specific impulse assuming 

frozen flow throughout. 

Answers: = 0.90; = 1.02; = 0.929; {Is)a = 250 sec. 



SYMBOLS 



A 

c 

Cp 

Cs 

C* 

Cf 

Cd 

d 

D 

F 

80 

h 

h 

J 

k 

L 

lit 

M 

m 

rii 

P 

R 

R' 

T 

V 

V 
vv 



area, m^(ft^) 

effective exhaust velocity, m/sec (ft/sec) 

specific heat at constant pressure, J/kg-K (Btu/lbm-°R) 

specific heat of solid, J/kg-K (Btu/lbm-°R) 

specific heat at constant volume, J/kg-K (Btu/lbm-°R) 

characteristic velocity, m/sec (ft/sec) 

thrust coefficient 

discharge coefficient (1/c*), sec/m (sec/ft) 
total derivative 
diameter, m (ft) 
thrust, N (Ibf) 

standard sea-level gravitational acceleration, 

9.8066 m/sec^(32.174 ft/sec^) 

enthalpy per unit mass, J/kg (Btu/lbm) (or altitude, km) 

specific impulse, sec or N-sec^/kg-m (Ibf-sec/lbm) 

mechanical equivalent of heat; J =4.186 J/cal in SI units or 

1 Btu = 777.9 ft-lbf 

specific heat ratio 

length of nozzle, m (ft) 

mass flow rate, kg/sec (Ibm/sec) 

mach number 

molecular mass, kg/kg-mol (or molecular weight, Ibm/lb-mol) 
molar fraction of species i 
pressure, N/m^(lbf/ft^ or Ibf/in.^) 

gas constant per unit weight, J/kg-K (ft-lbf/lbm-°R) {R = R' /dR) 

universal gas constant, 8314.3 J/kg mol-K (1544 ft-lb/lb mol-°R) 

absolute temperature, K (°R) 

velocity, m/sec (ft/sec) 

specific volume, m^/kg(ft^/lbm) 

propellant weight flow rate, N/sec (Ibf/sec) 



REFERENCES 101 



Greek Letters 

a 


half angle of divergent conical nozzle section 


P 


mass fraction of solid particles 


e 


nozzle expansion area ratio A2/A, 


^Cf 


thrust coefficient correction factor 


fc* 


c* correction factor 


Kd 


discharge correction factor 


Kf 


thrust correction factor 


Kv 


velocity correction factor 


X 


divergence angle correction factor for conical nozzle exit 


tin 


nozzle efficiency 



Subscripts 




a 


actual 


g 

i 


gas 

ideal, or a particular species in a mixture 


max 


maximum 


opt 


optimum nozzle expansion 


s 


solid 


sep 


point of separation 


t 


throat 


X 


any plane within rocket nozzle 


y 


any plane within rocket nozzle 


0 


stagnation or impact condition 


1 


nozzle inlet or chamber 


2 


nozzle exit 


3 


atmospheric or ambient 



REFERENCES 

3-1. A. H. Shapiro, The Dynamics and Thermodynamics of Compressible Fluid Flow, 
Vols. 1 and 2, Ronald Press Company, New York, 1953; and M. J. Zucrow and J. 
D. Hoffman, Gas Dynamics, Vols. I and II, John Wiley & Sons, New York, 1976 
(has section on nozzle analysis by method of characteristics). 

3-2. M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics , 
3rd ed., John Wiley & Sons, New York, 1996; also additional text, 1997. 

3-3. H. H. Koelle (Ed), Handbook of Astronautical Engineering, McGraw-Hill Book 
Company, New York, 1961. 

3-4. G. Hagemann, H. Immich, T. Nguyen, and G. E. Dummov, “Rocket Engine Nozzle 
Concepts,” Chapter 12 of Liquid Rocket Thrust Chambers: Aspects of Modeling, 
Analysis and Design, V. Yang, M. Habiballah, J. Hulka, and M. Popp (Eds), 
Progress in Astronautics and Aeronautics, Vol. 200, AIAA, 2004. 



102 



NOZZLE THEORY AND THERMODYNAMIC RELATIONS 



3-5. P. Vuillermoz, C. Weiland, G. Hagemann, B. Aupoix, H. Grosdemange, and M. 
Bigert, “Nozzle Design Optimization,” Chapter 13 of Liquid Rocket Thrust Cham- 
bers: Aspects of Modeling, Analysis and Design, V. Yang, M. Habiballah, J. Hulka, 
and M. Popp (Eds), Progress in Astronautics and Aeronautics, Vol. 200, AIAA, 
2004. 

3-6. “Liquid Rocket Engine Nozzles”, NASA SP-8120, 1976. 

3-7. G. P. Sutton, “Flow through a Combustion Zone,” Section of Chapter 3, Rocket 
Propulsion Elements, John Wiley & Sons, New York, 1956, 1963, and 1976. 

3-8. J. A. Muss, T. V. Nguyen, E. J. Reske, and D. M. McDaniels, “Altitude Compen- 
sating Nozzle Concepts for RLV,” AIAA Paper 97-3222, July 1997. 

3-9. G. V. R. Rao, “Recent Developments in Rocket Nozzle Configurations,” ARS Jour- 
nal, Vol. 31, No. 11, November 1961, pp. 1488-1494; and G. V. R. Rao, “Exhaust 
Nozzle Contour for Optimum Thrust,” Jet Propulsion, Vol. 28, June 1958, pp. 
377-382. 

3-10. J. M. Farley and C. E. Campbell, “Performance of Several Method-of- 
Characteristics Exhaust Nozzles,” NASA TN D-293 , October 1960. 

3-11. J. D. Hoffman, “Design of Compressed Truncated Perfect Nozzles,” Journal of 
Propulsion and Power, Vol. 3, No. 2, March-April 1987, pp. 150-156. 

3-12. G. P. Sutton, Stepped Nozzle, U.S. Patent 5,779,151, 1998. 

3-13. F. A. Williams, M. Barrere, and N. C. Huang, “Fundamental Aspects of Solid 
Propellant Rockets,” AGARDograph 116, Advisory Group for Aerospace Research 
and Development, NATO, October 1969, 783 pages. 

3-14. M. Barrere, A. Jaumotte, B. Fraeijs de Veubeke, and J. Vandenkerckhove, Rocket 
Propulsion, Elsevier Publishing Company, Amsterdam, 1960. 

3-15. R. N. Knauber, “Thrust Misalignments of Fixed Nozzle Solid Rocket Motors,” 
AIAA Paper 92-2873, 1992. 

3-16. J. S. Lilley, “The Design and Optimization of Propulsion Systems Employ- 
ing Scarfed Nozzles,” Journal of Spacecraft and Rockets, Vol. 23, No. 6, 
November -December 1986, pp. 597-604; and J. S. Lilley, “Experimental Val- 
idation of a Performance Model for Scarfed Nozzles,” Journal of Spacecraft and 
Rockets, Vol. 24, No. 5, September- October 1987, pp. 474-480. 

3-17. R. D. Zucker and O. Biblarz, Fundamentals of Gas Dynamics, 2nd ed., John Wiley 
& Sons, Hoboken, NJ, 2002. 



CHAPTER 4 



FLIGHT PERFORMANCE 



This chapter is an introduction to the flight performance of rocket-propelled vehi- 
cles such as spacecraft, space launch vehicles, missiles, or projectiles. It presents 
the subjects from a rocket propulsion point of view. Rocket propulsion systems 
provide forces to a flight vehicle and cause it to accelerate (or sometimes to 
decelerate), overcome drag forces, or change flight direction. Some propulsion 
systems provide also torques to the flight vehicles for rotation or maneuvers. The 
flight missions can be classified into several flight regimes: (1) flight within the 
earth’s atmosphere (air-to-surface missiles, surface-to-surface short-range mis- 
siles, surface-to-air missiles, air-to-air missiles, assisted takeoff units, sounding 
rockets, or aircraft rocket propulsion systems). Refs. 4-1 and 4-2; (2) near space 
environment (earth satellites, orbital space stations, or long-range ballistic mis- 
siles), Refs. 4-3 to 4-9; (3) lunar and planetary flights (with or without landing 
or earth return). Refs. 4-5 to 4-12; and (4) deep space exploration and sun 
escape. Each is discussed in this chapter. The chapter begins with the analy- 
sis of simple one-dimensional space flight and then treats more complex fight 
path scenarios and various flying rocket-propelled vehicles. The appendices give 
conversion factors, atmosphere properties, and a summary of key equations. 



4.1. GRAVITY-FREE DRAG-FREE SPACE FLIGHT 

This first simple rocket flight analysis applies to an outer space environment, 
far away from any star, where there is no air (thus no drag) and essentially no 
significant gravitational attraction. The flight direction is the same as the thrust 
direction (along the axis of the nozzle), namely, a one-dimensional, straight-line 
acceleration path; the propellant mass flow m and thus the propulsive thrust F 



103 



104 FLIGHT PERFORMANCE 



remain constant for the propellant burning duration tp. The thrust force F has 
been defined in Eq. 2-17. For a constant propellant flow the flow rate m is mpjtp, 
where nip is the total usable propellant mass. From Newton’s second law and for 
an instantaneous vehicle mass m and a vehicle flight velocity u 

F — m dujdt (4-1) 



For a rocket where the propellant flow rate is constant and the start or shut-off 
periods are very brief, the instantaneous mass of the vehicle m can be expressed 
as a function of the initial mass of the full vehicle mo, the propellant mass nip, 
the time at power cutoff tp, and the instantaneous time f. 



nip 

m = mo 1 = mo 

tp 



t^t\ 

mo tp ) 



= mo 




= mo 



1 - (1 -]V«)- 



(4-2) 

(4-3) 



Equation 4-3 expresses the vehicle mass in a form useful for trajectory cal- 
culations. The vehicle mass ratio IVR (final mass/initial mass or mf /mo ) and the 
propellant mass fraction ^ have been defined by Eqs. 2-7 and 2-8. They are 
related by 



f = 1 — IVR = 1 — m/ /mo = mp/mo (4-4) 

A definition of the various masses is shown in Fig. 4-1. The initial mass at 
takeoff mo equals the sum of the useful propellant mass nip plus the empty or 
final vehicle mass m^; my in turn equals the sum of the inert masses of the 
engine system (such as nozzles, tanks, cases, or unused, residual propellant) plus 
the guidance, control, electronics, and related equipment and the payload. The 
residual propellant remaining in the propulsion system after thrust termination 
is not available for combustion and is usually considered to be a part of the final 
engine mass after operation. This is the liquid propellant trapped in pipe pockets, 
valve cavities, and pumps or wetting the tank and pipe walls. For solid propellant 
rocket motors it is the remaining unbumed solid propellant, also called slivers, 
and sometimes also unbumed insulation. 

For constant propellant mass flow m and a finite propellant burning time t^ the 
total propellant mass mp is mtp and the instantaneous vehicle mass m = mo — liit. 
Equation 4-1 can be written as 

du = {F /m) dt = {cm/m)dt 

{cm)dt c{mp/tp)dt ct,/tp 

mo-mpt/tp moil - mpt/motp) l-^t/tp 

The start period and the shutdown period are very short and can be neglected. 
Integration leads to the maximum vehicle velocity at propellant burnout Up that 



4.1. GRAVITY-FREE DRAG-FREE SPACE FLIGHT 105 




FIGURE 4-1. Definitions of various vehicle masses. For solid propellant rocket motors 
the words “rocket engine, tanks, and structures” are to be replaced by “rocket motor 
nozzle, thermal insulation, and case.” 



can be attained in a gravity-free vacuum. When the initial flight velocity mq is 
not zero (mq 7 ^ 0) it is often called the velocity increment Am: 

Am = — cln(l — ^) -t- Mo = cln(mo/m/-) -F mq (4-5) 

If the initial velocity mq is assumed to be zero, then the velocity at thrust 
termination 

Up = Am = — cln(l — f) = — cln[mo/(mo — nip)] 

= — clnlV® = cln(l/]VR) 

= c \n{mo/mf) (4-6) 

The symbol In stands for the natural logarithm. This Up is the maximum veloc- 
ity increment Am that can be obtained in a gravity-free vacuum with constant 
propellant flow, starting from rest with mq = 0. The effect of variations in c, 7^, 
and ^ on the flight velocity increment is shown in Fig. 4-2. An alternate way to 
write Eq. 4-6 uses e, the base of the natural logarithm: 

= 1/IVR = mo/mf (4-7) 

The concept of the maximum attainable flight velocity increment Am in a 
gravity-free vacuum is useful in understanding the influence of the basic param- 
eters. It is used in comparing one propulsion system or vehicle with another. 



106 FLIGHT PERFORMANCE 



Specific impulse, sec 




Average effective exhaust velocity 




FIGURE 4-2. Maximum vehicle flight velocity in a gravitationless, drag-free space for 
different mass ratios and specific impulses (plot of Eq. 4-6). Single-stage vehicles can 
have values of IVR up to about 20 and multistage vehicles can exceed 200. 



one flight mission with another, or one proposed upgrade with another possible 
design improvement. 

From Eq. 4-6 it can be seen that the vehicle’s propellant mass fraction has a 
logarithmic effect on the vehicle velocity. By increasing this ratio from 0.80 to 
0.90, the interplanetary maximum vehicle velocity in gravitationless vacuum is 
increased by 43%. A mass fraction of 0.80 would indicate that only 20% of the 
total vehicle mass is available for structure, skin, payload, propulsion hardware, 



4.2. FORCES ACTING ON A VEHICLE IN THE ATMOSPHERE 107 



radios, guidance system, aerodynamic lifting surfaces, and so on; the remaining 
80% is useful propellant. It requires careful design to exceed 0.85; mass fraction 
ratios approaching 0.95 appear to be the probable practical limit for single-stage 
vehicles and currently known materials. When the mass fraction is 0.90, then 
i/m. = 0.1 and IVR = 10.0. This marked influence of mass fraction or mass 
ratio on the velocity at power cutoff, and therefore also the range, not only is 
true of interplanetary spaceships in a vacuum but applies to almost all types of 
rocket-powered vehicles. For this reason, importance is placed on saving inert 
mass on every vehicle component, including the propulsion system. 

Equation 4-6 can be modihed and solved for the effective propellant mass irip 
required to achieve a desired velocity increment for a given initial takeoff mass 
or a final burnout mass of the vehicle. The final mass consists of the payload, the 
structural mass of the vehicle, the empty propulsion system mass (which includes 
residual propellant), plus a small additional mass for guidance, communications, 
and control devices. Here mp = mo — m/: 

mp = - 1) = mo(l - (4-8) 

The flight velocity increment Up is proportional to the effective exhaust veloc- 
ity c and, therefore, to the specific impulse. Thus any improvement in Is (such as 
better propellants, more favorable nozzle area ratio, or higher chamber pressure) 
reflects itself in improved vehicle performance, provided that such an improve- 
ment does not also cause an excessive increase in rocket propulsion system inert 
mass, which causes a decrease in the effective propellant fraction. 



4.2. FORCES ACTING ON A VEHICLE IN THE ATMOSPHERE 

The external forces commonly acting on vehicles flying in the earth’s atmosphere 
are thrust, aerodynamic forces, and gravitational attractions. Other forces, such as 
wind or solar radiation pressure, are usually small and generally can be neglected 
for many simple calculations. 

The thrust is the force produced by the power plant, such as a propeller or a 
rocket. It usually acts in the direction of the axis of the power plant, that is, along 
the propeller shaft axis or the rocket nozzle axis. The thrust force of a rocket 
with constant mass flow has been expressed by Eq. 2-6 as a function of the 
effective exhaust velocity c and the propellant flow rate rh . In many rockets the 
mass rate of propellant consumption m is essentially constant, and the starting 
and stopping transients are usually very short and can be neglected. Therefore, 
the thrust is defined by Eq. 2-13 and is restated here with different parameters: 

F = cm = cmp/tp (4-9) 

As explained in Chapter 3, for a given propellant the value of the effective 
exhaust velocity c or specific impulse /j depends in part on the nozzle area 



108 FLIGHT PERFORMANCE 



ratio and the altitude. The value of c can increase by a relatively small factor 
of between 1.2 and 1.6 as altitude is increased with a maximim value in space 
(vacuum). 

The drag D is the aerodynamic force in a direction opposite to the flight path 
due to the resistance of the body to motion in a fluid. The lift L is the aerody- 
namic force acting in a direction normal to the flight path. They are expressed 
as functions of the flight speed m, the mass density of the fluid (air) in which the 
vehicle moves p, and a typical surface area A-. 



where Cl and Co are lift and drag coefficients, respectively. For airplanes and 
winged missiles the area A is the wing area. For wingless missiles or space launch 
vehicles it is the maximum cross-sectional area normal to the missile axis. The lift 
and drag coefficients are primarily functions of the vehicle configuration, flight 
Mach number, and angle of attack, which is the angle between the vehicle axis 
(or the wing plane) and the flight direction. For low flight speeds the effect of 
Mach number may be neglected, and the drag and lift coefficients are functions 
of the angle of attack. A typical variation of the drag and lift coefficients for a 
typical supersonic missile is shown in Fig. 4-3. The values of these coefficients 
reach a maximum near a Mach number of unity. For wingless vehicles the angle 
of attack a is usually very small (0 < a < 1°). The density and other properties 
of the atmosphere are listed in Appendix 2. The local density of the earth’s 
atmosphere can vary day by day by a factor up to 2 (for altitudes of 300 to 1200 
km) depending on solar activity and night-to-day temperature variations. This 
introduces a major unknown in the drag. The aerodynamic forces are affected 
by the flow and pressure distribution of the rocket exhaust gases, as explained in 
Chapter 20. 

The flight regime in the neighborhood of Mach 1 is called the transonic phase 
of flight. Flere strong unsteady aerodynamic forces can develop (due to buffeting) 
which are reflected in the steep rise and decrease of the coefficients seen in 
Fig. 4-3. Marginal transonic load capabilities have led to structural failures of 
flight vehicles. 

For space launch vehicles and ballistic missiles the integrated drag loss, when 
expressed in terms of Am, is typically 5 to 10% of the final ideal vehicle velocity. 
This relatively low value is due to the fact that the air density is low at high 
altitudes, when the velocity is high, and at low altitudes the air density is high 
but the flight velocity and thus the dynamic pressure are low. 

Gravitational attraction is exerted upon a flying space vehicle by all planets, 
stars, the moon, and the sun. Gravity forces pull the vehicle in the direction of 
the center of mass of the attracting body. Within the immediate vicinity of the 
earth, the attraction of other planets and bodies is negligibly small compared to 
the earth’s gravitational force. This force is the weight. 



L = CijpAu^ 
D = CojpAu^ 



(4-10) 

(4-11) 



4.2. FORCES ACTING ON A VEHICLE IN THE ATMOSPHERE 109 




0 1 2 3 4 5 6 

Mach number, M 




FIGURE 4-3. Variation of lift and drag coefficient with Mach number of the German 
V-2 missile based on body cross-sectional area with jet off and without exhaust plume 
effects at several angles of attack a. 



If the variation of gravity with the geographical features and the oblate shape 
of the earth are neglected, the acceleration of gravity varies inversely as the 
square of the distance from the earth’s center. If Rq is the mean radius of the 
spherical earth’s surface and go the acceleration on the earth’s surface at the 
earth’s effective radius Rq, the gravitational attraction g changes with altitude, 

g = goiRo/R?' 

= go[Ro/{Ro + h)f 



(4-12) 



110 FLIGHT PERFORMANCE 



where h is the altitude. At the equator the spherical earth’s radius is 6378.388 
km and the standard value of go is 9.80665 m/sec^. At a distance as far away 
as the moon, the earth’s gravity acceleration is only about 3.3 x 10“"^go- In n 
more accurate analysis the value of g will vary locally with the earth’s bulge at 
the equator, the high mountains, and the difference of densities of specific earth 
regions. 



4.3. BASIC RELATIONS OF MOTION 

For a vehicle that flies within the proximity of the earth, the gravitational attrac- 
tion of all other heavenly bodies may usually be neglected. Let it be assumed 
that the vehicle is moving in rectilinear equilibrium flight and that all control 
forces, lateral forces, and moments that tend to turn the vehicle are zero. The 
trajectory is two dimensional and is contained in a fixed plane. The vehicle has 
wings that are inclined to the flight path at an angle of attack a and that give a 
lift in a direction normal to the flight path. The direction of flight does not nec- 
essarily coincide with the direction of thrust. Figure 4-4 shows these conditions 
schematically. 

Let 0 be the angle of the flight path with the horizontal and \jr the angle 
of the direction of thrust with the horizontal. In the direction of the flight path 
the product of the mass and the acceleration has to equal the sum of all forces, 
namely the propulsive, aerodynamic, and gravitational forces: 

m{du/dt) = F cos(i/f — 9) — D — mg sin6> (4-13) 

The acceleration perpendicular to the flight path is u{d9/dt)\ for a constant value 
of u and the instantaneous radius R of the flight path it is u^/R. The equation of 
motion in a direction normal to the flight velocity is 

mu{d9 / dt) = F sin(i/f — 9) + L — mg cos 9 (4-14) 




FIGURE 4-4. Two-dimensional free-body force diagram for flying vehicle with wings 
and fins. 



4.3. BASIC RELATIONS OF MOTION 



111 



By substituting from Eqs. 4-10 and 4-11, these two basic equations can be 
solved for the accelerations as 

du F Cn T 

— = — cos(i/f — 0) pu^A — g sin 6 (4-15) 

dt m 2m 

dd F Cl 0 

u — = — sin(i/f — 0) H pu^A — g cos 0 (4-16) 

dt m 2m 

No general solution can be given to these equations, since tp, u, Co, Cl, p,0, or 
can vary independently with time, mission profile, or altitude. Also, Cl> and 
Cl are functions of velocity or Mach number. In a more sophisticated analysis 
other factors may be considered, such as the propellant used for nonpropulsive 
purposes (e.g., attitude control or flight stability). See Refs. 4-1, 4-8, 4-11, 
and 4-12 for a background of flight performance in some of the flight regimes. 
Different flight performance parameters are maximized or optimized for different 
rocket flight missions or flight regimes, such as An, range, orbit height and shape, 
time-to-target, or altitude. Rocket propulsion systems are usually tailored to fit 
specific flight missions. 

Equations 4-15 and 4-16 are general and can be further simplified for various 
special applications, as shown in subsequent sections. Results of such iterative 
calculations of velocity, altitude, or range using the above two basic equations 
often are adequate for rough design estimates. For actual trajectory analyses, nav- 
igation computation, space flight path determination, or missile-firing tables, this 
two-dimensional simplified theory does not permit sufficiently accurate results. 
The perturbation effects, such as those listed in Section 4.4, must then be con- 
sidered in addition to drag and gravity, and digital computers are necessary to 
handle the complex relations. An arbitrary division of the trajectory into small 
elements and a step-by-step or numerical integration to define a trajectory are 
usually indicated. The more generalized three-body theory includes the gravi- 
tational attraction among three masses (for example, the earth, the moon, and 
the space vehicle) and is considered necessary for many space flight problems 
(see Refs. 4-2 to 4-5). When the propellant flow and the thrust are not con- 
stant, when the flight path is three dimensional, the form and the solution to the 
equations above become more complex. 

A form of Eqs. 4-15 and 4-16 can also be used to determine the actual 
thrust or actual specific impulse during actual vehicle flights from accurately 
observed trajectory data, such as from optical or radar tracking data. The vehicle 
acceleration {du/dt) is essentially proportional to the net thrust and, by making 
an assumption or measurement on the propellant flow (which usually varies in a 
predetermined manner) and an analysis of aerodynamic forces, it is possible to 
determine the rocket propulsion system’s actual thrust under flight conditions. 

For each mission of flight one can obtain actual histories of velocities and 
distances traveled and thus complete trajectories when integrating Eqs. 4-15 and 
4-16. The more general case requires six equations; three for translation along 
each of three perpendicular axes and three for rotation about these axes. The 



112 FLIGHT PERFORMANCE 




FIGURE 4-5. Simplified free-body force diagram for vehicle without wings or fins. The 
force vector diagram shows the net force on the vehicle. 



choice of coordinate systems and the reference points can simplify the mathe- 
matical solutions (see Refs. 4-3 and 4-5). 

For a wingless rocket projectile, a space launch vehicle, or a missile with 
constant thmst and propellant flow, these equations can be simplified. In Fig. 4-5 
the flight direction 0 is the same as the thrust direction and lift forces for a 
symmetrical, wingless, stably flying vehicle can be assumed to be zero at zero 
angle of attack. For a two-dimensional trajectory in a single plane (no wind 
forces) and a stationary earth, the acceleration in the direction of flight is as 
follows: 



du 

dt 



hp 

1 - 



— g sin 0 



Co^pu^A/mo 
1 - Ct/tp 



(4-17) 



A force vector diagram in Fig. 4-5 shows the net force (by adding thrust, drag, 
and gravity vectors) to be at an angle to the flight path, which will be curved. 
These types of diagram form the basis for iterative trajectory numerical solutions. 

The relationships in this section are for a two-dimensional flight path, one that 
lies in a single plane. If maneuvers out of that plane are also made (e.g., due 
to solar attraction, thmst misalignment, or wind), then the flight paths become 
three dimensional and another set of equations will be needed to describe these 
flights. Reference 4-1 describes equations for the motion of rocket projectiles 
in the atmosphere in three dimensions. It requires energy and forces to push a 
vehicle out of its flight plane. Trajectories have to be calculated accurately in 
order to reach the intended flight objective and today all are done with the aid of 
a computer. A good number of computer programs for analyzing flight trajecto- 
ries exist and are maintained by aerospace companies or government agencies. 



4.3. BASIC RELATIONS OF MOTION 113 



Some are two dimensional, relatively simple, and used for making preliminary 
estimates or comparisons of alternative flight paths, alternative vehicle designs, 
or alternative propulsion schemes. Several use a stationary flat earth, while others 
use a rotating curved earth. Three-dimensional programs also exist, are used for 
more accurate flight path analyses, and include some or all significant perturba- 
tions, orbit plane changes, or flying at angles of attack. As explained in Ref. 4-4, 
they are more complex. 

If the flight trajectory is vertical (as for a sounding rocket), then sin0 =1.0 
and COS0 = 0. Equation 4-17 can be modified: 

du cKitp CdIpu^A/mq 

— = — Q — (4— lo) 

dt 1 - );t/tp 1 - 



The velocity at the end of burning can be found by integrating between the 
limits of f = 0 and t = tp when u = uq and u = Up. The first two terms can 
readily be integrated. The last term is of significance only if the vehicle spends 
a considerable portion of its time within the atmosphere. It can be integrated 
graphically or by numerical methods, and its value can be designated as BCoA/niQ 
such that 





1 - t,tltp 



dt 



The cutoff velocity or velocity at the end of propellant burning Up is then 



BCdA 

Up = -c\n{\ - l;) - gtp hMo (4-19) 

mo 

where mq is the initial velocity, such as may be given by a booster, g is an 
average gravitational attraction evaluated with respect to time and altitude from 
Eq. 4-12, and c is a time average of the effective exhaust velocity, which is a 
function of altitude. 

There are always a number of trade-offs in selecting the best trajectory for a 
rocket projectile. Eor example, for a fixed thrust there is a trade-off between burn- 
ing time, drag, payload, maximum velocity, and maximum altitude (or range). 
Reference 4-2 describes the trade-offs between payload, maximum altitude, and 
flight stability for a sounding rocket. 

If aerodynamic forces outside the earth’s atmosphere are neglected (operate 
in a vacuum) and no booster or means for attaining an initial velocity (mq = 0) is 
assumed, the velocity at the end of the burning reached in a vertically ascending 
trajectory will be 



Up = -cln(l - O - gtp 
= —c In IVR — 'gtp 
= cln(l/lV«) - gtp 



(4-20) 



114 FLIGHT PERFORMANCE 



The first term on the right side is usually the largest and is identical to Eq. 4-6. 
It is directly proportional to the effective rocket exhaust velocity and is very 
sensitive to changes in the mass ratio. The second term is related to the earth’s 
gravity and is always negative during ascent, but its magnitude is small if the 
burning time tp is short or if the flight takes place in high orbits or in space 
where g is comparatively small. 

For a flight that is not following a vertical path, the gravity loss is a function of 
the angle between the flight direction and the local horizontal; more specifically, 
the gravity loss is the integral of g sin 9 dt. 

For the simplified two-dimensional case the net acceleration a for vertical 
takeoff at sea level is 



where a /go is the initial takeojf acceleration in multiples of the sea-level grav- 
itational acceleration go, and Fo/wo is the thrust- to- weight ratio at takeoff. For 
large surface-launched vehicles, this initial-thrust-to-initial-weight ratio typically 
has values between 1.2 and 2.2; for small missiles (air-to-air, air-to-surface, and 
surface-to-air types) this ratio is usually larger, sometimes even as high as 50 or 
100. The final or terminal acceleration af of a vehicle in vertical ascent usually 
occurs just before the rocket engine is shut off and/or before the propellant is 
completely consumed. If drag is neglected, then 



The second version of this equation applies if the powered flight path traverses 
a substantial change in altitude and thus a change in the value of g . In a gravity- 
free environment this equation becomes a/- /go = Fj/wf. In rockets with constant 
propellant flow the final acceleration is usually also the maximum acceleration, 
because the vehicle mass to be accelerated has its minimum value just before 
propellant exhaustion, and for ascending rockets the thrust usually increases with 
altitude. If this terminal acceleration is too large (and causes overstressing of the 
structure, thus necessitating an increase in structure mass), then the thrust can be 
designed to a lower value for the last portion of the burning period. In manned 
flights the maximum acceleration is limited to the maximum g loading that can 
be withstood by the crew. 

Example 4-1. A simple single-stage rocket for a rescue flare has the following charac- 
teristics. Its flight path nomenclature is shown in the accompanying sketch. 



a = {Fogo/wo) - go 
a/go = (fo/wo) - 1 



(4-21) 

(4-22) 



«//^o = (Ff/wf) - 1 

= iFf/mf)go - g /go 



(4-23) 



4.3. BASIC RELATIONS OF MOTION 115 



Launch weight 

Useful propellant weight 

Effective specific impulse 

Launch angle 0 (relative to horizontal) 

Bum time tp (with constant thrust) 



4.0 Ibf 
0.4 Ibf 
120 sec 
80° 



1.0 sec 




The heavy line in the ascending trajectory 
designates the powered portion of the 
flight. 



Drag may be neglected since the flight velocities are low. Assume that the acceleration 
of gravity is unchanged from its sea-level value go, which then makes the propellant mass 
numerically equal to the propellant weight in the EE system, or 0.4 Ibm. Also assume 
that start and stop transients are short and can be ignored. 

Solve for the initial and final acceleration of powered flight, the maximum trajectory 
height and the time to reach maximum height, the range or horizontal distance to impact, 
and the angle at propulsion cutoff. 

SOLUTION. We divide the flight path into three portions: the powered flight for 1 sec, 
the unpowered ascent after cutoff, and the free-fall descent. The thrust is obtained from 
Eq. 2-5: 



The initial accelerations along the x and y horizontal and vertical directions are, from 
Eq. 4-22, 

(ao)y = go[(T sin0/w) - 1] = 32.17[(48/4.0) sin 80° - 1] = 348 ft/sec^ 

(ao)x = go{Fcose/w) = 32.17(48/4.0) cos 80° = 67.03 ft/sec^ 

At thrust termination the initial flight acceleration becomes 



The vertical and horizontal components of the velocity Up at the end of powered flight are 
obtained from Eq. 4-20. Note that the vehicle mass has been diminished by the propellant 
that has been consumed: 

{Up)y = cln(mo//ny) sin0 — g^tp = 32.17 x 120 x ln(4/3.6) x 0.984 — 32.17 x 1.0 
= 368 m/sec 

(Up)^ = c In (mo /my) cost? = 32.17 x 120 x ln(4/3.6) x 0.1736 = 70.6 m/sec 



F = hw/tp = 120 X 0.4/1.0 = 48 Ibf 




116 FLIGHT PERFORMANCE 



The trajectory angle with the horizontal at rocket cutoff for a dragless flight is 



The final acceleration is found, using Eq. 4-22 with the final mass, as 
Of = 400 mlsec^. 

For the powered flight, the coordinates at propulsion burnout and Xp can be calcu- 
lated from the time integration of their respective velocities. The results are 

yp = ctp[\ - \n{mQ/mf)l{mQlnif - 1)] sin0 - lIlgQtj 

= 32.17120[1 - ln(4/3.6)/(4/3.6 - 1)] x 0.984 - i x 32.17 x (1.0)^ 

= 181 ft 

Xp = ctp\\ — \n{mQ / nif) / {triQ / trif — l)]cosfl 

= 32.17120[1 - ln(4/3.6)/(4/3.6 - 1)] x 0.173 
= 34.7 ft 

The unpowered part of the trajectory reaches zero vertical velocity at its zenith. The height 
gained in unpowered free flight may be obtained by equating the vertical kinetic energy 
at power cutoff to its equivalent potential energy. 



The maximum height or zenith location thus becomes yz = 2105 -|- 181 = 2259 ft. 
What remains now is to solve the free-flight portion of vertical descent. The time for 
descent from the zenith is tz = -s/^y , /go = 11-85 sec and the final vertical or impact 
vertical velocity (m/)j — goE = 381 ft/sec. 

The total horizontal range to the zenith is the sum of the powered and free-flight 
contributions. During free flight the horizontal velocity remains unchanged at 70.6 ft/sec 
because there are no accelerations (i.e., no drag, wind, or gravity component). We now 
need to find the free-flight time from burnout to the zenith, which is r = (Up)y/go = 
11.4 sec. The total free-flight time becomes tff = 11.4-1- 11.85 = 23.25 sec. 

Now, the horizontal or total range becomes Ajc = 34.7 -|- 70.6 x 23.25 = 1676 ft. 

The impact angle would be around 79°. If drag had been included, solving this problem 
would have required information on the drag coefficient (Co) and a numerical solution 
using Eq. 4-18. All resulting velocities and distances would turn out somewhat lower in 
value. A set of flight trajectories for sounding rockets has been worked out in Ref. 4-3. 



4.4. SPACE FLIGHT 

Newton’s law of gravitation defines the attraction of gravitational force Fg 
between two bodies in space as follows: 



tan-' (368/70.6) = 79.1° 



so that 



go(yz - yp) 

iyz -yp) = \(Up)]lg, = 1(368)2/32.17 = 2105 ft 




Fg = Gm\m2/R^ = fxm2/R^ 



(4-24) 



4.4. SPACE FLIGHT 117 



Here G is the universal gravity constant (G = 6.670 x 10“"m^/kg-sec^),mi 
and m 2 are the masses of the two attracting bodies (such as the earth and the 
moon, the earth and a spacecraft, or the sun and a planet), and R is the distance 
between their centers of mass. The earth’s gravitational constant /x is the product 
of Newton’s universal constant G and the mass of the earth, mi (5.974 x 10^’^kg). 
It is /X = 3.98600 x lO'^^m^/sec^. 

The rocket offers a means for escaping the earth for lunar and interplanetary 
travel, for escaping our solar system, and for creating a stationary or moving 
station in space. The flight velocity required to escape from the earth can be 
found by equating the kinetic energy of a moving body to the work necessary to 
overcome gravity, neglecting the rotation of the earth and the attraction of other 
celestial bodies: 

imw^ = m j g dR 

By substituting for g from Eq. 4-12 and by neglecting air friction the follow- 
ing relation for the escape velocity Ve is obtained: 



Ve = Ro 



2g0 

Rq 4- h 




(4-25) 



Here Rq is the effective earth mean radius (6374.2 km), h is the orbit altitude 
above sea level, and go is the acceleration of gravity at the earth surface (9.806 
m/sec). The satellite flight radius R measured from the earth’s center is R = 
Ro + h. The velocity of escape at the earth’s surface is 11,179 m/sec or 36,676 
ft/sec and does not vary appreciably within the earth’s atmosphere, as shown 
by Fig. 4-6. Escape velocities for surface launch are given in Table 4-1 for 
the sun, the planets, and the moon. Launching from the earth’s surface at escape 
velocity is not practical. As a vehicle ascends through the earth’s atmosphere, it is 
subject to severe aerodynamic heating and dynamic pressures. A practical launch 
vehicle has to traverse the atmosphere at relatively low velocity and accelerate to 
the high velocities beyond the dense atmosphere. For example, during a portion 
of the Space Shuttle’s ascent, its main engines are actually throttled to a lower 
thrust to avoid excessive pressure and heating. Alternatively, an escape vehicle 
can be launched from an orbiting space station or from an orbiting Space Shuttle. 

A rocket spaceship can become a satellite of the earth and revolve around the 
earth in a fashion similar to that of the moon. Satellite orbits are usually ellip- 
tical and some are circular. Low earth orbits, typically below 500 km altitude, 
are designated by the letters LEO. The altitude of the orbit is usually above the 
earth’s atmosphere, because this minimizes the expending of energy to overcome 
the small drag which pulls the vehicle closer to the earth. The effects of the radi- 
ation in the Van Allen belt on human beings and sensitive equipment sometimes 
necessitate the selection of an earth orbit at low altitude. 

For a circular trajectory the velocity of a satellite must be sufficiently high so 
that its centrifugal force balances the earth’s gravitational attraction: 

mu^ /R — mg 



118 FLIGHT PERFORMANCE 



15,000 



10,000 



5,000 




Altitude, 10® ft 



5 10 

Altitude, 10® m 



50 



40 



t3 

O 

0> 

Q_ 




c. 

UJ 



FIGURE 4-6. Orbital energy, orbital velocity, period of revolution, and earth escape 
velocity of a space vehicle as a function of altitude for circular satellite orhits. It is based 
on a spherical earth and neglects the earth’s oblate shape, rotation, and atmospheric drag. 



For a circular orbit, the satellite velocity Us is found by using Eq. 4-12, 

Mi = RoV go/ (Ro + h) = II /R (4-26) 

which is smaller than the escape velocity by a factor of -Jl. The period t in 
seconds of one revolution for a circular orbit relative to a stationary earth is 

T = l7i{Ro + h)/us = 27t(7?o + hfl^KRo^ ) (4-27) 



The energy E necessary to bring a unit of mass into a circular satellite orbit 
neglecting drag consists of kinetic and potential energy, namely, 





1 

= -Rogo 



Ro + 2/f 
Ro + h 



(4-28) 



The escape velocity, satellite velocity, satellite period, and satellite orbital energy 
are shown as functions of altitude in Fig. 4-6. 

A satellite circulating around the earth at an altitude of 300 miles or 482.8 km 
has a velocity of about 7375 m/sec or 24,200 ft/sec, circles a stationary earth in 
1.63 hr, and ideally requires an energy of 3.35 x 10^ J to place 1 kg of spaceship 
mass into its orbit. An equatorial satellite in a circular orbit at an altitude of 



^ ^ I ? 

fZ) o 3 C 

m S 
> 



c ^ 

O >» D ^ 
.ti o fCP 
cd > ciH o 
h U 1^ 

i-H ^ V2 

"o O on ~> 

H ^ 

y o ca ^ 



'5 > 

D g 
Oh rn 
00 ^ 



T3 

O 

cn 

’c 

(D 

> 

a 

<D 

K 



-3 

Id 

Q 



*c 



c3 

x: 

U 



QQ 



c3 ^ 

13 
oC 



X 

t: 

C3 

W 



c 



T3 -t= ^ 
cd X _ 
Qi C 

O O 

^ t-i-^ ’~~! 

g o ^ 



C3 

z 



o 


o 


o 


o 


o^ 


o 


O 


o 


o 


o 


o 


o 


00 


o 


o 




o 


o 


o 


o 


o 


o 


o 


CO 


CM 


CO 






I> 






o^ 


o 


m5 


CM 




o' 


wS 


M3 


Os 


vo" 


CM 




o” 














VO 


CO 


CN 


CO 




M3 























'sO G^ 

00 o r-- o ^ cN 

^WO'0^00^>0^0^0^^ 



CO 

I> 

CNl 



m 00 o^ m ^ — 

CNj — 



-H Tj- CN uo CO 0^ 

'^COOOCOUOONCO^r^ 

-^coiriinwoco-HO-H 





o 


M3 

O 


M3 

00 


o 

p 


VO 




CM 


p 


CM 


o 

o^ 


o 

I/O 


o 


d 


d 




d 


00 


VO 

o^ 


VO 


I> 


d 



o 


VO 


o 


O 


CM 


O 


O 


o 


o 


o 


o 


o 


r- 


I> 


o 




M3 


o 


o 


o 


o 


VO 


p. 




M3 






I> 








I> 


o^ 


co*^ 

o^ 


CO 




CM 


CM 


M3 


CO*' 


CM 


l> 


o” 

in 


VO 



r" 


Vi 


!Z3 


c/3 


c/3 


c/3 


Lh 


u 


Vh 


;-l 


u 


1 1 I « 


>» 






?o 


>, 


cd 


c3 


cd 


C3 


c3 


o .2 


C3 




Kj 


C3 


cd 


D 


D 


(U 


(3 


D 


"3 


T3 


T3 


03 


03 


03 




>* 




?o 


>* 


o 0 =: 


I 




















•c 2 


I 






M3 














oi 




O 


o 


VO 


00 


M3 


M3 








CO 


0\ 




CM 


Os 


p 




p 


oo 


p 


l> 






VO 


M3 




Os 


of 


of 


Of 




CM 


00 


CM 


M3 


00 


1-H 


CM 


00 




00 








CM 


CO 


M3 










CM 



CO 

00 



CO 


p 




M3 




p 










d 


i> 


00 


Os 




I> 


M3 


Os 


VO 


Os 




VO 


o 


of 


CM 


I> 


00 


M3 


r-- 


Os 






1-^ 




CM 


I> 


of 


00 


of 


00 
















CM 


of 


VO 



g o 

goo 

5 s s 



C3 

w 



g § 

I— S c/5 hJ 



c 

H.2 
O 3 
Z S 



I 

TD 

C 



I 

^ ; 



Qi X 



Cu 

c 



r5 W 



119 



120 FLIGHT PERFORMANCE 



6.611 earth radii (about 26,200 miles, 42,200 km, or 22,700 nautical miles) has a 
period of revolution of exactly 24 hr. It will appear stationary to an observer on 
earth. This is known as a synchronous satellite in geo synchronous earth orbit, 
usually abbreviated as GEO. It is used extensively for communications satellite 
applications. In Section 4.7 on launch vehicles we will describe how the payload 
of a given space vehicle diminishes as the orbit circular altitude is increased 
and as the inclination (angle between orbit plane and earth equatorial plane) is 
changed. See Refs. 4-3, 4-4, 4-5, 4-6, and 4-9. 



Elliptical Orbits 

The circular orbit described above is a special case of the more general elliptic 
orbit shown in Fig. 4-7; here the earth (or any other heavenly body around which 
another body is moving) is located at one of the focal points of this ellipse. The 
equations of motion may be derived from Kepler’s laws, and the elliptical orbit 
can be described as follows, when expressed in polar coordinates: 




(4-29) 



where u is the velocity of the body in the elliptical orbit, R is the instantaneous 
radius from the center of the attracting body (a vector quantity, which changes 
direction as well as magnitude), a is the major axis of the ellipse, and /r is the 
earth’s gravitational constant with a value of 3.986 x lO'^ m^/sec^. The symbols 
are dehned in Fig. 4-7. From this equation it can be seen that the velocity Up is 
a maximum when the moving body comes closest to its focal point at the orbit’s 
perigee and that its velocity is a minimum at its apogee. By substituting for R 




FIGURE 4-7. Elliptical orbit; the attracting body is at one of the focal points of the 
ellipse. 



4.4. SPACE FLIGHT 121 



in Eq. 4-29, and by defining the ellipse’s shape factor e as the eccentricity of the 
ellipse, e = — b^/a, the apogee and perigee velocities can be expressed as 



Uq 



Ub 



/^(l - e) 
a{\ + e) 



M(1 +e) 

a{\ — e) 



(4-30) 

(4-31) 



Another property of an elliptical orbit is that the product of velocity and instan- 
taneous radius remains constant for any location x or y on the ellipse, namely, 
— UyRy = uR. The exact path that a satellite takes depends on the velocity 
(magnitude and vector orientation) with which it is started or injected into its 
orbit. 

For interplanetary transfers the ideal mission can be achieved with minimum 
energy in a simple transfer ellipse, as suggested originally by Hohmann (see Ref. 
4-6). Assuming the planetary orbits about the sun to be circular and coplanar, 
it can be demonstrated that the path of minimum energy is an ellipse tangent 
to the planetary orbits as shown in Fig. 4-8. This operation requires a velocity 
increment (relatively high thrast) at the initiation (planet A at fi) and another at 
termination (planet B at t 2 )'- both increments are the velocity differences between 
the respective circular planetary velocities and the perigee and apogee velocities 
which define the transfer ellipse. The thrust levels at the beginning and end 
maneuvers of the Hohmann ellipse must be high enough to give a short operating 
time and the acceleration of at least 0.01 go, but preferably more. With electrical 
propulsion these accelerations would be about 10“^go> the operating time would 
be weeks or months, and the best transfer trajectories would be very different 
from a Hohmann ellipse; they are described in Chapter 17. 



Planet B at 




FIGURE 4-8. Schematic diagram of interplanetary transfer paths. These same transfer 
maneuvers apply when going from a low-altitude earth satellite orbit to a higher orbit. 



122 FLIGHT PERFORMANCE 

The departure date or the relative positions of the launch planet and the target 
planet for a planetary transfer mission are critical, because the spacecraft has 
to meet with the target planet when it arrives at the target orbit. The transfer 
time (?2 — fi) for a Hohmann ellipse flight starting on earth is about 116 hr to 
go to the moon and about 259 days to Mars. If a faster orbit (shorter transfer 
time) is desired (see dashed lines in Fig. 4-8), it requires more energy than a 
Hohmann transfer ellipse. This means a larger vehicle with more propellant and 
a larger propulsion system or more total impulse. There also is a time window 
for a launch of a spacecraft that will make a successful rendezvous. For a Mars 
mission an earth-launched spacecraft may have a launch time window of more 
than two months. A Hohmann transfer ellipse or a faster transfer path applies 
not only to planetary flight but also to earth satellites, when an earth satellite 
goes from one circular orbit to another (but within the same plane). Also, if 
one spacecraft goes to a rendezvous with another spacecraft in a different orbit, 
the two spacecraft have to be in the proper predetermined positions prior to the 
launch to simultaneously reach their rendezvous location. 

When the launch orbit (or launch planet) is not in the same plane as the target 
orbit, then additional energy will be needed by applying thrust in a direction 
normal to the launch orbit plane. More information is in Refs. 4-3, 4-4, 4-6, 
and 4-10. 

Example 4-2. A satellite is launched from a circular equatorial parking orbit at an 
altitude of 160 km into a coplanar circular synchronous orbit by using a Hohmann transfer 
ellipse. Assume a homogeneous spherical earth with a radius of 637 1 km. Determine the 
velocity increments for entering the transfer ellipse and for achieving the synchronous 
orbit at 42,200 km altitude. See Fig. 4-8 for the terminology of the orbits. 

SOLUTION. The orbits are Ra = 6.531 x 10'’ m; Rb = 48.571 x 10® m. The major axis 
a of the transfer ellipse is 



Qte = + Rb) = 27.551 x 10® m 



The orbit velocities of the two satellites in the two orbits are 

UA = fp/RA = [3.986005 x 10*V6.571 x 10®]*''^ = 7788 m/sec 
ub = f pIRb = 2864.7 m/sec 

The velocities needed to enter and exit the transfer ellipse are 

{u,,)a = Vm[(2/7?a) - (1/fl)]'''^ = 10,337 m/sec 

(m/c)b = Vm[(2/7^b) - (1/a)]'''^ = 1394 m/sec 

The changes in velocity going from parking orbit to ellipse and from ellipse to final 
orbit are 



tsuA = \(ute)A — ua\ = 2549 m/sec 
Aub = \ub — (ute)B = 1471 m/sec 



4.4. SPACE FLIGHT 123 



Local Elliptical ballistic 




FIGURE 4-9. Long-range ballistic missiles follow an elliptical free flight trajectory, 
which is drag free, with the earth’s center as one of the focal points. The surface launch 
is usually vertically up (not shown here) but the flight path is quickly tilted during the early 
powered flight to enter into an elliptic trajectory. The hallistic range is the arc distance 
on the earth’s surface. The same elliptical flight path can be used by launch vehicles for 
satellites; another powered flight period occurs (called orhit injection) just as the vehicle 
is at its elliptical apogee (as indicated hy the arrow), causing the vehicle to enter an orhit. 



The total velocity change for the transfer maneuvers is 



Awtotai = Ama + Amb = 4020 m/sec 



Figure 4-9 shows the elliptical transfer trajectory of a ballistic missile or a satellite 
ascent vehicle. During the initial powered flight the trajectory angle is adjusted hy the guid- 
ance system to an angle that will allow the vehicle to reach the apogee of its elliptical path 
exactly at the desired orhit altitude. For the ideal satellite the simplified theory assumes 
that orbit injection is an essentially instantaneous application of the total impulse as the 
ballistic trajectory reaches its apogee or zenith. In reality the rocket propulsion system 
operates over a finite time, during which gravity losses and changes in altitude occur. 



Deep Space 

Lunar and interplanetary missions include circumnavigation, landing, and return 
flights to the moon, Venus, Mars, and other planets. The energy necessary to 



124 FLIGHT PERFORMANCE 



Operation of retro rocket 
to slow vehicle down 
to satellite velocity (1-5%) 



C 03 S* let 

trajec 




. . at touchdo 

Vertical launch and 



Retro-rock 




Acceleration 



maneuver to attain 



Retro-rocket maneuver 
to de-orbit into a landing 
approach (1-5%) 



interplanetary 
orbit (1-10%) 



Coast in orbit 



FIGURE 4-10. Schematic diagram of typical powered flight maneuvers during a hypo- 
thetical two-dimensional interplanetary mission with a landing. The numbers indicate 
typical thrust magnitudes of the maneuvers in percent of launch takeoff thrust. This is not 
drawn to scale. Heavy lines show powered flight path segments. 

escape from the earth can be calculated as jtnv^ from Eq. 4-25. It is 6.26 x 
10^ J/kg, which is more than that required for a satellite. The gravitational attrac- 
tion of various heavenly bodies and their respective escape velocities depends 
on their masses and diameters; approximate values are listed in Table 4-1. An 
idealized diagram of an interplanetary landing mission is shown in Fig. 4-10. 

The escape from the solar system requires approximately 5.03 x 10^ J/kg. 
This is eight times as much energy as is required for escape from the earth. 
There is technology to send small, unmanned probes away from the sun to outer 
space; as yet there needs to be an invention and demonstrated proof of a long- 
duration, novel, rocket propulsion system before a mission to the nearest star can 
be achieved. The trajectory for a spacecraft to escape from the sun is either a 
parabola (minimum energy) or a hyperbola. See Refs. 4-6 and 4-10. 

Perturbations 

This section gives a brief discussion of the disturbing torques and forces which 
cause perturbations or deviations from any intended space flight path or any 
satellite’s flight orbit. For a more detailed treatment of flight paths and their 
perturbations, see Refs. 4-3, 4-4, and 4-13. A system is needed to measure 
the satellite’s position and deviation from the intended flight path, to determine 
the needed periodic correction maneuver, and then to counteract, control, and 
correct them. This is called orbit maintenance ; it corrects the perturbed or altered 
orbit by periodically applying small rocket propulsion forces in predetermined 
directions. Typically, the corrections are performed by a set of small reaction 
control thrusters which provide predetermined total impulses into the desired 



4.4. SPACE FLIGHT 



125 




FIGURE 4-11. The regression of nodes is shown as a rotation of the plane of the orbit. 
The direction of the movement will be opposite to the east-west components of the 
earth’s satellite motion. 



directions. These corrections are needed throughout the life of the spacecraft (for 
1 to 20 years or sometimes more) to overcome the effects of the disturbances 
and maintain the intended flight regime. 

Perturbations can be categorized as short term and long term. The daily or 
orbital period oscillating forces are called diurnal and those with long periods 
are called secular. 

High-altitude earth satellites (36,000 km and higher) experience perturbing 
forces primarily as gravitational pull from the sun and the moon, with the forces 
acting in different directions as the satellite flies around the earth. This third-body 
effect can increase or decrease the velocity magnitude and change its direction. 
In extreme cases the satellite can come very close to the third body, such as a 
planet or one of its moons, and undergo what is called a hyperbolic maneuver 
that will radically change the trajectory. This encounter can be used to increase 
or decrease the energy of the satellite and intentionally change the velocity and 
the shape of the orbit. 

Medium- and low-altitude satellites (500 to 35,000 km) experience perturba- 
tions because of the earth’s oblateness. The earth bulges in the vicinity of the 
equator and a cross section through the poles is not entirely circular. Depending 
on the inclination of the orbital plane to the earth equator and the altitude of the 
satellite orbit, two perturbations result: (1) the regression of the nodes and (2) 
shifting of the apsides line (major axis). Regression of the nodes is shown in 
Fig. 4-11 as a rotation of the plane of the orbit in space, and it can be as high 
as 9° per day at relatively low altitudes. Theoretically, regression does not occur 
in equatorial orbits. 

Figure 4-12 shows an exaggerated shift of the apsidal line, with the center of 
the earth remaining as a focus point. This perturbation may be visualized as the 



126 



FLIGHT PERFORMANCE 




FIGURE 4-12. Shifting of the apsidal line of an elliptic orbit from position 1 to 2 
because of the oblateness of the earth. 



movement of the prescribed elliptical orbit in a fixed plane. Obviously, both the 
apogee and perigee points change in position, the rate of change being a function 
of the satellite altitude and plane inclination angle. At an apogee altitude of 1000 
nautical miles (n.m.) and a perigee of 100 n.m. in an equatorial orbit, the apsidal 
drift is approximately 10° per day. 

Satellites of modern design, with irregular shapes due to protruding antennas, 
solar arrays, or other asymmetrical appendages, experience torques and forces 
that tend to perturb the satellite’s position and orbit throughout its orbital life. 
The principal torques and forces result from the following factors: 

1 . Aerodynamic drag . This factor is significant at orbital altitudes below 500 
km and is usually assumed to cease at 800 km above the earth. Reference 
4-8 gives a detailed discussion of aerodynamic drag, which, in addition 
to affecting the attitude of unsymmetrical vehicles, causes a change in 
elliptical orbits known as apsidal drift, a decrease in the major axis, and a 
decrease in eccentricity of orbits about the earth. See Refs. 4-6, -8, -12, 
and -13. 

2. Solar radiation. This factor dominates at high altitudes (above 800 km) 
and is due to impingement of solar photons upon satellite surfaces. The 
solar radiation pressure p (N/m^) on a given surface of the satellite in the 
vicinity of the earth exposed to the sun can be determined as 

p = 4.5x 10“®cos6»[(l -k,)cos6> + 0.67krf] (4-32) 

where 9 is the angle (degrees) between the incident radiation vector and the 
normal to the surface and and k^i are the specular and diffuse coefficients 
of reflectivity. Typical values are 0.9 and 0.5, respectively, for ks and kd 
on the body and antenna and 0.25 and 0.01, respectively, for kg and kd 
with solar array surfaces. The radiation intensity varies as the square of 



4.4. SPACE FLIGHT 127 



the distance from the sun (see Refs. 4-4 and 4-14). The torque T on the 
vehicle is given by T = pAl, where A is the projected area and I is the 
offset distance between the spacecraft’s center of gravity and the center 
of solar pressure. For a nonsymmetrical satellite with a large solar panel 
on one side the radiation will cause a small torque, which will rotate the 
vehicle. 

3. Gravity gradients. Gravitational torque in spacecraft results from a vari- 
ation in the gravitational force on the distributed mass of a spacecraft. 
Determination of this torque requires knowledge of the gravitational field 
and the distribution of spacecraft mass. This torque decreases as a function 
of the orbit radius and increases with the offset distances of masses within 
the spacecraft (including booms and appendages); it is most significant in 
large spacecraft or space stations operating in relatively low orbits (see 
Refs. 4-4 and 4-15. 

4. Magnetic field . The earth’s magnetic field and any magnetic moment within 
the satellite interact to produce torque. The earth’s magnetic field precesses 
about the earth’s axis but is very weak (0.63 and 0.31 gauss at poles and 
equator, respectively). This field is continually fluctuating in direction and 
intensity because of magnetic storms and other influences. Since the field 
strength decreases with \/R^ with the orbital altitude, magnetic field forces 
are often neglected in the preliminary analysis of satellite flight paths (see 
Ref. 4-16). 

5. Internal accelerations . Deployment of solar array panels, the shifting of 
liquid propellant, the movement of astronauts or other masses within the 
satellite, or the “unloading” of reaction wheels can produce torques and 
forces. 

6. For precise low earth orbits the oblateness of the earth (diameter at equator 
is slighty lerger than diameter between poles), high mountains, or earth 
surface areas of different densities will perturb these orbits. 

We can categorize satellite propulsion needs according to function as listed in 
Table 4-2, which shows the total impulse “budget” applicable to a typical high- 
altitude, elliptic orbit satellite. The control system designer often distinguishes 
two different kinds of station-keeping orbit corrections needed to keep the satellite 
in a synchronous position. The east-west correction refers to a correction that 
moves the point at which a satellite orbit intersects the earth’s equatorial plane 
in an east or west direction; it usually corrects forces caused largely by the 
oblateness of the earth. The north- south correction counteracts forces usually 
connected with the third-body effects of the sun and the moon. 

In many satellite missions the gradual changes in orbit caused by perturbation 
forces are not of concern. However, in certain missions it is necessary to com- 
pensate for these perturbing forces and maintain the satellite in a specific orbit 
and in a particular position in that orbit. For example, a synchronous communi- 
cations satellite in a Geo-Synchronous Earth Orbit or GEO needs to maintain its 
position and its orbit, so it will be able to (1) keep covering a specific area of the 



128 FLIGHT PERFORMANCE 



TABLE 4-2. Propulsion Functions and Total Impulse Needs of a 
2000-lbm Geosynchronous Satellite with a 7-Year Life 



Function 


Total Impulse 
(N-sec) 


Acquisition of orbit 


20,000 


Attitude control (rotation) 


4,000 


Station keeping, E-W 


13,000 


Station keeping, N-S 


270,000 


Repositioning (Am, 200 ft/sec) 


53,000 


Control apsidal drift (third-body attraction) 


445,000 


Deorbit 


12,700 


Total 


817,700 



earth or communicate with the same stations on earth within its line of sight and 
(2) not become a hazard to other satellites in this densely occupied synchronous 
equatorial orhit. Another example is a Low Earth Orbit or LEO communications 
satellite system with several coordinated satellites; here at least one satellite has 
to be in a position to receive and transmit radio-frequency (RE) signals to spe- 
cific points on the earth. Their orbits, and the relative positions of these several 
satellites with respect to each other, need to be controlled and maintained (see 
Ref. 4-3). 

Orbit maintenance means applying small correcting forces and torques peri- 
odically to compensate for perturbation effects; for GEO it is typically every few 
months. Typical velocity increments for the orbit maintenance of synchronous 
satellites require a Am between 10 and 50 m/sec per year. For a satellite mass of 
about 2000 kg a 50-m/sec correction for a 10-year orbit life would need a total 
impulse of about 100,000 N-sec, which corresponds to a propellant mass of 400 
to 500 kg (about a quarter of the satellite mass) if done by a small monopro- 
pellant or bipropellant thrust. It would require much less propellant if electrical 
propulsion were used, but in some spacecraft the inert mass of the power supply 
would increase. See Refs. 4-6, 4-13, and 4-14. 

Mission Velocity 

A convenient way to describe the magnitude of the energy requirement of a 
space mission is to use the concept of the mission velocity. It is the sum of 
all the flight velocity increments needed (in all the vehicle’s stages) to attain 
the mission objective even though these increments are provided by different 
propulsion systems. In the simplified sketch of a planetary landing mission of 
Fig. 4-10, it is the sum of all the Am velocity increments shown by the heavy 
lines (rocket-powered flight segments) of the trajectories. Even through some 
of the velocity increments were achieved by retro-action (a negative propulsion 
force to decelerate the flight velocity), these maneuvers required energy and their 
absolute magnitude is counted in the mission velocity. The initial velocity from 



4.4. SPACE FLIGHT 129 



TABLE 4-3. Typical Estimated Space Shuttle Incremental Flight Velocity Breakdown 



Ideal satellite velocity 


7790 m/sec 


Am to overcome gravity losses 


1220 m/sec 


Am to turn the flight path from the vertical 


360 m/sec 


Am to counteract aerodynamic drag 


118 m/sec 


Orbit injection 


145 m/sec 


Deorbit maneuver to re-enter atmosphere and aerodynamic braking 


60 m/sec 


Correction maneuvers and velocity adjustments 


62 m/sec 


Initial velocity provided by the earth’s rotation at 28.5° latitude 


—408 m/sec 


Total required mission velocity 


9347 m/sec 



the earth’s rotation (464 m/sec at the equator and 408 m/sec at a launch station at 
28.5° latitude) does not have to be provided by the vehicle’s propulsion systems. 
For example, the required mission velocity for launching at Cape Kennedy, bring- 
ing the space vehicle into an orbit at 110 km, staying in orbit for a while, and 
then entering a deorbit maneuver has the Am components shown in Table 4-3. 

The required mission velocity is the sum of the absolute values of all trans- 
lation velocity increments that have forces going through the center of gravity 
of the vehicle (including turning maneuvers) during the flight of the mission. It 
is the theoretical hypothetical velocity that can be attained by the vehicle in a 
gravity-free vacuum, if all the propulsive energy of the momentum- adding thrust 
chambers in all stages were to be applied in the same direction. It is useful 
for comparing one flight vehicle design with another and as an indicator of the 
mission energy. 

The required mission velocity has to be equal to the “supplied” mission veloc- 
ity, that is, the sum of all the velocity increments provided by the propulsion 
systems of each of the various vehicle stages. The total velocity increment to be 
“supplied” by the shuttle’s propulsion systems for the shuttle mission described 
below (solid rocket motor strap-on boosters, main engines and, for orbit injec- 
tion, also the increment from the orbital maneuvering system — all shown in 
Fig. 1-14) has to equal or exceed 9347 m/sec. If reaction control system propel- 
lant and an uncertainty factor are added, it will need to exceed 9621 m/sec. With 
chemical propulsion systems and a single stage, we can achieve a space mission 
velocity of 4000 to 13,000 m/sec, depending on the payload, mass ratio, vehicle 
design, and propellant. With two stages it can be between perhaps 12,000 and 
22,000 m/sec. 

Rotational maneuvers, described later, do not change the flight velocity and 
some analysts do not add them to the mission velocity requirements. Also, main- 
taining a satellite in orbit against long-term perturbing forces (see prior section) 
is often not counted as part of the mission velocity. However, the designers need 
to provide additional propulsion capability and propellants for these purposes. 
These are often separate propulsion systems, called reaction control systems. 

Typical vehicle velocities required for various interplanetary missions have 
been estimated as shown in Table 4-4. By starting interplanetary journeys from 



130 FLIGHT PERFORMANCE 



TABLE 4-4. Approximate Vehicle Mission Velocities for Typical Interplanetary 
Missions 



Mission 


Ideal Velocity 
(km/sec) 


Approximate 
Actual Velocity 
(km/sec) 


Satellite orbit around earth (no return) 


7.9-10 


9.1-12.5 


Escape from earth (no return) 


11.2 


12.9 


Escape from moon 


2.3 


2.6 


Earth to moon (soft landing on moon, no 


13.1 


15.2 


return) 


Earth to Mars (soft landing) 


17.5 


20 


Earth to Venus (soft landing) 


22 


25 


Earth to moon (landing on moon and return to 


15.9 


17.7 


earth®) 


Earth to Mars (landing on Mars and return to 


22.9 


27 


earth®) 


“As.sumes air braking witbin atmospheres. 







TABLE 4-5. Approximate Relative Payload-Mission Comparison Chart for Typical 
High-Energy Chemical Multistage Rocket Vehicles 


Mission 


Relative Payload® (%) 


Earth satellite 


100 


Earth escape 


35-45 


Earth 24-hr orbit 


10-25 


Moon landing (hard) 


35-45 


Moon landing (soft) 


10-20 


Moon circumnavigation (single fly-by) 


30-42 


Moon satellite 


20-30 


Moon landing and return 


1-4 


Moon satellte and return 


8-15 


Mars flyby 


20-30 


Mars satellite 


10-18 


Mars landing 


0.5-3 



“300 nautical miles (555.6 km) earth orbit is 100% reference. 



a space satellite station, a considerable saving in this vehicle velocity can be 
achieved, namely, the velocity necessary to achieve the earth-circling satellite 
orbit. As the space flight objective becomes more ambitious, the mission velocity 
is increased. For a given single or multistage vehicle it is possible to increase the 
vehicle’s terminal velocity, but usually only at the expense of payload. Table 4-5 
shows some typical ranges of payload values for a given multistage vehicle as 
a percentage of a payload for a relatively simple earth orbit. Thus a vehicle 
capable of putting a substantial payload into a near-earth orbit can only land a 



4.5. FLIGHT MANEUVERS 131 



very small fraction of this payload on the moon, since it has to have additional 
upper stages, which displace payload mass. Therefore, much larger vehicles are 
required for space flights with high mission velocities if compared to a vehicle 
of less mission velocity but identical payload. The values listed in Tables 4-4 
and 4-5 are only approximate because they depend on specific vehicle design 
features, the propellants used, exact knowledge of the trajectory -time relation, 
and other factors that are beyond the scope of this short treatment. 



4.5. FLIGHT MANEUVERS 

In this section we describe different flight maneuvers and relate them to specific 
propulsion system types. See Refs. 4-5, 4-11, 4-12, 4-13, and 4-14. The three 
categories of maneuvers are: 

1. In translation maneuvers the rocket propulsion thrust vector goes through 
the center of gravity of the vehicle. The vehicle momentum is changed 
in the direction of the flight velocity. An example of several powered 
(translational maneuvers) and unpowered (coasting) segments of a complex 
space flight trajectory is shown in schematic, simplified form in Fig. 4-10. 
To date, most maneuvers have used chemical propulsion systems. 

2. In truly rotational maneuvers there is no net thrust acting on the vehicle. 
These are true force couples that apply only torque. It requires four thrusters 
to be able to rotate the vehicle in either direction about any one axis (two 
thrusters apart, firing simultaneously, but in opposite rotational directions). 
These types of maneuver are usually provided by reaction control systems. 
Most have used multiple liquid propellant thrusters, but in recent years 
selected space missions have used electrical propulsion. 

3. A combination of categories 1 and 2, such as a misaligned thrust vector 
of a large thmst chamber that does not go exactly through the center of 
gravity of the vehicle. The misalignment can be corrected by changing 
the vector direction of the main propulsion system (thrust vector control) 
during powered flight or by applying a simultaneous compensating torque 
from a separate reaction control system. 

The following types of space flight maneuvers and vehicle accelerations use 
rocket propulsion. All propulsion operations are usually controlled (started, mon- 
itored, and stopped) by the vehicle’s guidance and control system. 

a. First stage, its upper stage propulsion systems and strap-on boosters add 
momentum during launch and ascent. They require rocket propulsion of 
high or medium thrusts and limited durations (typically 0.7 to 8 min). To 
date all have used chemical propulsion systems. They constitute the major 
mass of the space vehicle and are discussed further in the next section. 



132 FLIGHT PERFORMANCE 



b. Orbit injection or transferring from one orbit to another requires accurately 
predetermined total impulses. It can be performed by the main propulsion 
system of the top stage of the launch vehicle. More often it is done by a 
separate propulsion system at lower thrust levels than the upper stages in 
item (a) above. Orbit injection can be a single thrust operation after ascent 
from an earth launch station. If the flight path is a Hohmann transfer ellipse 
(minimum energy) or a faster transfer orbit, then two thrust application 
periods are necessary, one at the beginning and one at the end of the 
transfer path. For injection into earth orbit, the thrust levels are typically 
between 200 and 45,000 N or 50 and 1 1,000 Ibf, depending on the payload 
size, transfer time, and specific orbit. If the new orbit is higher, then the 
thrusts are applied in the flight direction. If the new orbit is at a lower 
altitude, then the thrusts must be applied in a direction opposite to the 
flight velocity vector. The transfer orbits can also be achieved with a very 
low thrust level (0.001 to 1 N) using an electric propulsion system, but 
the flight paths will be very different (multiloop spiral) and the transfer 
duration will be much longer. This is explained in Chapter 17. Similar 
maneuvers are also performed with lunar or interplanetary flight missions, 
like the planetary landing mission shown schematically in Fig. 4-10. 

c. Velocity vector adjustment and minor in-flight correction maneuvers are 
usually performed with low-thrust, short-duration, and intermittent (puls- 
ing) operations, using a reaction control system with multiple small liquid 
propellant thrusters, both for translation and rotation. The vernier rockets 
on a ballistic missile are used to accurately calibrate the terminal velocity 
vector for improved target accuracy. The reaction control rocket systems 
in a space launch vehicle will allow accurate orbit injection adjustment 
maneuvers after it is placed into orbit by another, less accurate propulsion 
system. Midcourse guidance-directed correction maneuvers for the trajecto- 
ries of deep space vehicles fall also into this category. Propulsion systems 
for orbit maintenance maneuvers , also called station-keeping maneuvers 
(to overcome perturbing forces), keep a spacecraft in its intended orbit and 
orbital position and are also considered to be part of this category. 

d. Reentry and landing maneuvers can take several forms. If the landing 
occurs on a planet that has an atmosphere, then the drag of the atmosphere 
will slow down the reentering vehicle. For a multiple elliptical orbit the 
drag will progressively reduce the perigee altitude and the perigee velocity 
on every orbit. Landing at a precise, preplanned location requires a particu- 
lar velocity vector at a predetermined altitude and distance from the landing 
site. The vehicle has to be rotated into the right position and orientation, 
so as to use its heat shield correctly. The precise velocity magnitude and 
direction prior to entering the denser atmosphere are critical for minimiz- 
ing the heat transfer (usually to the vehicle’s heat shield) and to achieve 
touchdown at the intended landing site or, in the case of ballistic missiles, 
the intended target. This usually requires a relatively minor maneuver (low 
total impulse). If there is very little or no atmosphere (for instance, landing 



4.5. FLIGHT MANEUVERS 133 



on the moon or Mercury), then a reverse thrust has to be applied during 
descent and touchdown. The rocket propulsion system usually has variable 
thrust to assure a soft landing and to compensate for the decrease in vehi- 
cle mass as propellant is consumed during descent. The U.S. lunar landing 
rocket engine, for example, had a 10 to 1 thrust variation. 

e. Rendezvous and docking involve both rotational and translational maneu- 
vers of small reaction control thrusters. Rendezvous and its time windows 
were discussed on page xxx. Docking (sometimes called lockon) is the 
linking up of two spacecraft and requires a gradual gentle approach (low 
thrust, pulsing node thrusters) so as not to damage the spacecraft. 

f. Simple rotational maneuvers rotate the vehicle on command into a specific 
angular position so as to orient or point a telescope, instrument, solar panel, 
or antenna for purposes of observation, navigation, communication, or solar 
power reception. Such a maneuver is also used to keep the orientation of 
a satellite in a specific direction; for example, if an antenna needs to be 
continuously pointed at the center of the earth, then the satellite needs to 
be rotated around its own axis once every satellite revolution. Rotation 
is also used to point a nozzle of the primary propulsion system into its 
intended direction just prior to its start. It can also provide pulsed thrust 
for achieving flight stability, or for correcting angular oscillations, that 
would otherwise increase drag or cause tumbling of the vehicle. Spinning 
or rolling a vehicle about its axis will improve flight stability but will 
also average out the misalignment in a thrust vector. If the rotation needs 
to be performed quickly, then a chemical multithruster reaction control 
system is used. If the rotational changes can be done over a long period of 
time, then an electrical propulsion system with multiple thrusters is often 
preferred. 

g. A change of plane of the flight trajectory requires the application of a thrust 
force (through the vehicle center of gravity) in a direction normal to the 
original plane of the flight path. This is usually performed by a propulsion 
system that has been rotated (by the reaction control system) into the proper 
nozzle orientation. This maneuver is done to change the plane of a satellite 
orbit or when going to a planet, such as Mars, whose orbit is inclined to 
the plane of the earth’s orbit. 

h. Deorbiting and disposal of used or spent spacecraft is required today to 
remove space debris. The spent spacecraft should not become a hazard to 
other spacecraft. A relatively small thmst will cause the vehicle to go to 
a low enough elliptical orbit so that atmospheric drag will cause further 
slowing. In the dense regions of the atmosphere the reentering, expended 
vehicle will typically break up or overheat (bum up). 

i. Emergency or alternative mission. If there is a malfunction in a spacecraft 
and it is decided to abort the mission, such as a premature quick return to 
the earth without pursuing the originally intended mission, then some of 
the rocket engines can be used for an alternate mission. For example, the 



134 FLIGHT PERFORMANCE 



main rocket engine in the Apollo lunar mission service module is normally 
used for retroaction to attain a lunar orbit and for return from lunar orbit 
to the earth; it can be used for emergency separation of the payload from 
the launch vehicle and for unusual midcourse corrections during translunar 
coast, enabling an emergency earth return. 

Table 4-6 lists the maneuvers that have just been described, together with 
some others, and shows the various types of rocket propulsion system (as men- 
tioned in Chapter 1) that have been used for each of these maneuvers. The items 
with a double mark “x x” have been the preferred methods in recent years. The 
table omits several propulsion systems, such as solar thermal or nuclear rocket 
propulsion, because these have not yet flown in a routine space mission. The 
three propulsion systems on the right of Table 4-6 are electrical propulsion sys- 
tems and they have very high specific impulse (see Table 2-1), which makes 
them very attractive for deep space missions and for certain station-keeping jobs 
(orbit maintenance). However, they can be applied only to missions with suffi- 
ciently long thrust action time for reaching the desired vehicle velocity or rotation 
positions with very small acceleration. 



Reaction Control System 

The functions of a reaction control system have been described in the previous 
section on flight maneuvers. They are used for the maneuvers identified by para- 
graphs c, e, and g. In some vehicle designs they are also used for tasks described 
in b, part of d, and f, if the thrust levels are low. 

A reaction control system (RCS), often called an auxiliary rocket propulsion 
system, is needed to provide for trajectory corrections (small Am additions) as 
well as correcting the rotational or attitude position of almost all spacecraft and all 
major launch vehicles. If only rotational maneuvers are made, it has been called 
an attitude control system. The nomenclature has not been consistent throughout 
the industry or the literature. 

An RCS can be incorporated into the payload stage and each of the stages of 
a multiple-stage vehicle. In some missions and designs the RCS is built into only 
the uppermost stage; it operates throughout the flight and provides the control 
torques and forces for all the stages. For large vehicle stages the thrust level 
of multiple thrusters of an RCS can be large (500 to 15,000 Ibf) and for termi- 
nal stages in small satellites they can be small (0.01 to 10.0 Ibf) and they can 
be pulsed as often as commanded by the vehicle flight control system. Liquid 
propellant rocket engines with multiple thrusters have been used for almost all 
launch vehicles and the majority of all spacecraft. Cold gas systems were used 
with early spacecraft design. In the last decade an increasing number of elec- 
trical propulsion systems have been used, primarily on spacecraft, as described 
in Chapter 17. The life of an RCS may be short (when used on an individual 
vehicle stage), or it may see use throughout the mission duration (perhaps more 
than 10 years) when part of an orbiting spacecraft. 



4.5. FLIGHT MANEUVERS 135 



TABLE 4-6. Types of Rocket Propulsion System Commonly Used for Different Flight 
Maneuvers or Application 





Liquid Propellant 
Rocket Engines 


Solid Propellant 
Rocket Motors 


Electrical 

Propulsion 


\ Propulsion 

\ System 

Flight 

Maneuvers and \ 

Applications j, \ 


High Thrust, Liquid Propellant 
Rocket Engine, with Turbopump 

Medium to Low Thrust, Liquid 
Propellant Rocket Engine 

Pulsing Liquid Propellant, Multiple 
Small Thrusters 


Large Solid Propellant Rocket 
Motor, Often Segmented 

Medium to Small Solid Propellant 
Motors 


Arc Jet, Resisto Jet 

Ion Propulsion, Electromagnetic 
Propulsion 

Pulsed Plasma Jet 


Launch vehicle booster 


X X 


X X 




Strap-on motor/engine 


X X 


X X 




Upper stages of launch 


XX XX 


X XX 




vehicle 








Satellite orbit injection and 


X X 


X X 


X X 


transfer orbits 








Flight velocity adjustments, 


X XX 




X X 


flight path corrections. 








orbit changes 








Orbit/position maintenance, 


X X 




XXX 


rotation of spacecraft 








Docking of two spacecraft 


X X 






Reentry and landing. 


X X 






emergency maneuvers 








Deorbit 


X X 


X 


X 


Deep space, sun escape 


X X 




X 


Tactical missiles 




X X 




Strategic missiles 


XXX 


X X 




Missile defense 


X 


XX XX 





Legend : x = in use : x x = preferred for use. 



The vehicle attitude has to he controlled about three mutually perpendicular 
axes, each with two degrees of freedom (clockwise and counterclockwise rota- 
tion), giving a total of six degrees of rotational freedom. Pitch control raises or 
lowers the nose of the vehicle, yaw torques induce a motion to the right or the 
left side, and roll torques will rotate the vehicle about its axis, either clockwise 
or counterclockwise. In order to apply a true torque it is necessary to use two 
thrust chambers of exactly equal thrust and equal start and stop times, placed an 
equal distance from the center of mass. Figure 4-13 shows a simple spherical 



136 FLIGHT PERFORMANCE 




y 



FIGURE 4-13. Simplified attitude control system diagram for spacecraft. It requires 
12 thrusters (identified as x, y, z) to allow the application of pure torques about three 
perpendicular axes. The four unlabeled thrusters are needed for translation maneuvers 
along the z axis. They are shown here in four clusters. 



spacecraft attitude control system; thrusters x — x' or x' — x apply torques that 
rotate about the X axis. There is a minimum of 12 thrusters in this system, but 
some spacecraft with geometrical or other limitations on the placement of these 
nozzles or with provisions for redundancy may actually have more than 12. The 
same system can, by operating a different set of nozzles, also provide transla- 
tion forces; for example, if one each of the thrust units x and x' were operated 
simultaneously, the resulting forces would propel the vehicle in the direction of 
the Y axis. With clever design it is possible to use fewer thrusters, but they will 
usually not provide a pure torque. 

An RCS usually contains the following major subsystems: (1) sensing devices 
for determining the attitude, velocity, and position of the vehicle with respect to a 
reference direction at any one time, such as provided by gyroscopes, star-trackers, 
or radio beacons; (2) a control-command system that compares the actual space 
and rotary position with the desired or programmed position and issues command 
signals to change the vehicle position within a desired time period; and (3) devices 
for changing the angular position, such as a set of high-speed gyroscopic wheels 
or a set of attitude control thrust-providing devices. See Refs. 4-13 and 4-14. 



4.6. EFFECT OF PROPULSION SYSTEM ON VEHICLE PERFORMANCE 137 



A precise attitude angular correction can also be achieved by the use of an 
inertial or high-speed rotating reaction wheel, which applies torque when its 
rotational speed is increased or decreased. While these wheels are quite simple 
and effective, the total angular momentum change they can supply is generally 
small. By using a pair of supplementary attitude control thrust rocket units it 
is possible to unload or respin each wheel so it can continue to supply small 
angular position corrections as needed. 

The torque T of a pair of thrust chambers of thrust F and a separation distance 
I is applied to give the vehicle with an angular or rotational moment of inertia 
Ma an angular acceleration of magnitude a: 



T = Fl= Mc,a (4-33) 

For a cylinder of equally distributed mass Ma = and for a homogeneous 
sphere it is Ma = jmr . The largest possible practical value of moment arm I 
will minimize the thrust and propellant requirements. If the angular acceleration 
is constant over a time period t, the vehicle will move at an angular speed co and 
through a displacement angle 9, namely 

CO = at and 9 = ^at^ (4-34) 

Commonly a control system senses a small angular disturbance and then com- 
mands an appropriate correction. For this detection of an angular position change 
by an accurate sensor it is actually necessary for the vehicle to undergo a slight 
angular displacement. Care must be taken to avoid overcorrection and hunting 
of the vehicle position or the control system. This is one of the reasons many 
spacecraft require extremely short multiple pulses (0.010 to 0.030 sec) and low 
thrust (0.01 to 100 N) (see Refs. 4-11, 4-13, and 4-14). 

Reaction control systems can be characterized by the magnitude of the total 
impulse, the number, thrust level, and direction of the thrusters, and their duty 
cycles. The duty cycle refers to the number of thrust pulses, their operating times, 
the times between thrust applications, and the timing of these short operations 
during the mission operating period. For a particular thruster, a 30% duty cycle 
means an average active cumulative thrust period of 30% during the propul- 
sion system’s flight duration. These propulsion parameters can be determined 
from the mission, the guidance and control approach, the desired accuracy, flight 
stability, the likely thrust misalignments of the main propulsion systems, the 
three-dimensional flight path variations, the perturbations to the trajectory, and 
several other factors. Some of these parameters are often difficult to determine. 



4.6. EFFECT OF PROPULSION SYSTEM ON VEHICLE PERFORMANCE 

This section gives several methods for improving flight vehicle performance. 
Most of these enhancements, listed below, are directly influenced by the flight 



138 FLIGHT PERFORMANCE 



mission and by the selection or design of the propulsion system. A few of the 
flight vehicle performance improvements do not depend on the propulsion system. 
Most of those listed below apply to all missions, but some are peculiar to some 
missions only. 

1 . The ejfective exhaust velocity c and the specific impulse Is usually have a 
direct effect on the vehicle’s flight performance. The vehicle’s final velocity 
increment Am can be inceased by a higher This can be done by using 
a more energetic chemical propellant (see Chapters 7 and 12), by a higher 
chamber pressure, and, for upper stages operating at high altitudes, also 
by a larger nozzle area ratio. Electrical propulsion (higher fi) can enhance 
vehicle performance, but, as explained later, the very low thrusts do limit 
this type to certain space missions. See Chapter 17. 

2. The mass ratio mo/m/ has a logarithmic effect. It can be increased in several 
ways. One way is by reducing the final mass mf, which consists of the inert 
hardware plus the nonusable, residual propellant mass. Reducing the inert 
mass implies lighter structures, smaller payloads, lighter guidance/control 
devices, or less unavailable residual propellant; this means going to stronger 
structural materials at higher stresses, more efficient power supplies, or 
smaller electronic packages. During design there is always great emphasis 
to reduce all hardware masses and the residual propellants to their practical 
minima. Another way is to increase the initial vehicle mass, and use a higher 
thrust and more propellant, but with a smaller increase in the structure or 
inert propulsion system masses. 

3. Reducing the burning time (i.e., increasing the thrust level) will reduce the 
gravitational loss in some applications. However, the higher acceleration 
usually requires more structural and propulsion system mass, which in turn 
causes the mass ratio to be less favorable. 

4. The drag, which can be considered as a negative thrust, can be reduced 
in at least four ways. The drag has several components: (a) The form drag 
depends on the aerodynamic shape. A slender pointed nose or sharp, thin 
leading edges of fins or wings have less drag than a stubby, blunt shape, 
(b) A vehicle with a small cross-sectional area has less drag. A propulsion 
design that can be packaged in a long, thin shape will be preferred, (c) The 
drag is proportional to the cross-sectional or frontal vehicle area. A higher 
propellant density will decrease the propellant volume and therefore will 
allow a smaller cross section, (d) The skin drag is caused by the friction of 
the air flowing over all the vehicle’s outer surfaces. A smooth contour and 
a polished surface are usually better. The skin drag is also influenced by 
the propellant density, because it gives a smaller volume and thus a lower 
surface area, (e) The base drag is the fourth component; it is a function of 
the local ambient air pressure acting over the surface of the vehicle’s base 
or bottom plate. It is influenced by the nozzle exit design (exit pressure), 
the discharge of turbine exhaust gases, and the geometry of the vehicle 
base design. It is discussed further in Chapter 20. 



4.7. FLIGHT VEHICLES 139 



5. The length of the propulsion nozzle often is a significant part of the overall 
vehicle or stage length. As was described in Chapter 3, there is an optimum 
nozzle contour and length, which can be determined by trade-off analysis. 
A shorter nozzle length or multiple nozzles on the same propulsion system 
allow a somewhat shorter vehicle; on many designs this implies a somewhat 
lighter vehicle structure and a slightly better vehicle mass ratio. 

6. The final vehicle velocity at propulsion termination can be increased by 
increasing the initial velocity uq. By launching a satellite in an eastward 
direction the rotational speed of the earth is added to the final satellite 
orbital velocity. This tangential velocity of the earth is about 464 m/sec or 
1523 ft/sec at the equator and the Sea Launch from a ship on the equator 
takes full advantage of this velocity increment. For an easterly launch at 
John F. Kennedy Space Center (latitude of 28.5° north) this extra velocity 
is about 408 m/sec or 1340 ft/sec. Conversely, a westerly satellite launch 
has a negative initial velocity and thus requires a higher-velocity increment. 
Another way to increase u is to launch a spacecraft from a satellite or an 
aircraft, which increases the initial vehicle velocity and allows launching 
in the desired direction, or to launch an air-to-surface missile from an 
airplane. An example is the Pegasus three-stage space launch vehicle, which 
is launched from an airplane. 

7. For vehicles that fly in the atmosphere it is possible to increase the range 
when aerodynamic lift is used to counteract gravity and reduce gravity 
losses. Using a set of wings or flying at an angle of attack increases the 
lift, but it also increases the drag. This lift can also be used to increase the 
maneuverability and trajectory flexibility. 

8. When the flight velocity u is close to the rocket’s effective exhaust velocity 
c, the propulsive efficiency is the highest (Eq. 2-23) and more of the 
rocket exhaust gas energy is transformed into the vehicle’s flight energy. 
Trajectories where u is close in value to c for a major portion of the flight 
therefore need less propellant. 

Several of these influencing parameters can be optimized. Therefore, for every 
mission or flight application there is an optimum propulsion system design and 
the propulsion parameters that define the optimum condition are dependent on 
vehicle or flight parameters. 



4.7. FLIGHT VEHICLES 

As mentioned, the vast majority of rocket-propelled vehicles are simple, single 
stage, and use solid propellant rocket motors. Most are used in military applica- 
tions, as described in the next section. This section discusses more sophisticated 
multistage space launch vehicles and mentions others, such as large ballistic mis- 
siles (often called strategic missiles) and some sounding rockets. All have some 



140 FLIGHT PERFORMANCE 



intelligence in their guidance and navigation system. The total number of multi- 
stage rocket vehicles produced worldwide in the last few years has been between 
100 and 220 per year. 

A single stage to orbit (LEO) is limited in the payload it can carry. Figure 4-2 
shows that a high-performance single-stage vehicle with a propellant fraction of 
0.95 and an average 4 of 400 sec can achieve an ideal terminal velocity of about 
12,000 m/sec without payload. If the analysis includes drag and gravity forces, a 
somewhat higher value of 4, maneuvers in the trajectory, and an attitude control 
system, it is likely that the payload would be between 0.2 and 1 .4% of the gross 
takeoff mass, depending on the design. For a larger percentage of payload, and 
for ambitious missions, we use vehicles with two or more stages as described 
here. 

Multistage Vehicles 

Multistep or multistage rocket vehicles permit higher vehicle velocities, more 
payload for space vehicles, and improved performance for long-range ballistic 
missiles. After the useful propellant is fully consumed in a particular stage, 
the remaining empty mass of that expended stage is dropped from the vehicle 
and the operation of the propulsion system of the next step or stage is started. 
The last or top stage, which is usually the smallest, carries the payload. The 
empty mass of the expended stage or step is separated from the remainder of 
the vehicle, because it avoids the expenditure of additional energy for further 
accelerating a useless mass. As the number of steps is increased, the initial 
takeoff mass can be decreased; but the gain in a smaller initial mass becomes less 
apparent when the total number of steps is large. Actually, the number of steps 
chosen should not be too large, because the physical mechanisms become more 
numerous, complex, and heavy. The most economical number of steps is usually 
between two and six, depending on the mission. Several different multistage 
launch vehicle configurations have been used successfully and four are shown in 
Fig. 4-14. Most are launched vertically, but a few have been launched from an 
airplane, such as the three-stage Pegasus space vehicle. See Example 4-3. 

The payload of a multistage rocket is essentially proportional to the takeoff 
mass, even though the payload is only a very small portion of the initial mass. If 
a payload of 50 kg requires a 6000-kg multistage rocket, a 500-kg payload would 
require a 60,000-kg rocket unit with an identical number of stages and a similar 
configuration with the same payload fraction and the same propellants. When the 
operation of the upper stage is started, immediately after thrust termination of 
the lower stage, then the total ideal velocity of a multistage vehicle of tandem 
or series-stage arrangement is simply the sum of the individual stage velocity 
increments. For n stages in series (one on top of each other) the final velocity 
increment Auf is 



n 




(4-35) 



4.7. FLIGHT VEHICLES 141 




Sustainer stage 
(contains propellant 
for booster thrust) 



Second 

stage 



First stage 
or booster 



/ 




/— Dropable booster 
\ engine ring package 
\ (without propellant) 




Partial staging 



Sustainer 





Parallel staging Piggy-back 

staging 



FIGURE 4-14. Simplified schematic sketches of four geometric configurations for 
assembling individual stages into a launch vehicle. 



The individual velocity increments are given by Eq. 4-6. For the simplified case 
of a vacuum flight in a gravity-free field this can be expressed as 

Am/- = Cl ln(l /IVRi) -|- C 2 ln(l/]VR 2 ) + C 3 ln(l/]V« 3 ) -f ■ ■ ■ (4-36) 

This equation defines the maximum velocity an ideal tandem multistage vehi- 
cle can attain in a gravity-free vacuum environment. For more accurate actual 
trajectories the individual velocity increments can be determined by integrating 
Eqs. 4-15 and 4-16, which consider drag and gravity losses. Other losses or 
trajectory perturbations can also be included, as mentioned earlier in this chapter. 
Such an approach requires numerical solutions. 

For two- or three-stage vehicles the overall vehicle mass ratio (initial mass at 
takeoff to final mass of last stage) can reach values of over 100 (corresponding 




142 FLIGHT PERFORMANCE 



to an equivalent single-stage propellant mass fraction ^ of 0.99). Figure 4-2 
can be thus divided into regions for single- and tandem multistage vehicles. 
Equation 4-36 does not apply to parallel or partial staging as identified in 
Fig. 4-14. For stages where more than one propulsion system is operating at 
the same time and producing thrust in the same direction, the effective specific 
impulse and nozzle exhaust velocity is given by Eqs. 10-12 to 10-14. 

The first sketch in Fig. 4-14 depicts a very common configuration and the 
stages are stacked vertically on top of each other, as in the Minuteman long- 
range missile or the Russian Zenit (Zenith) launch vehicle.* Partial staging was 
used on the early versions of the U.S. Atlas vehicle. It allows all engines to 
be started at launching, thus avoiding an altitude start of the sustainer engine, 
which was unknown in those early days. Fiquid propellant rocket engines can 
be shut off on the launch stand if a failure is sensed prior to lift-off. The two 
Atlas booster engines arranged in a doughnut-shaped assembly are dropped off 
in flight. The third sketch has two or more separate booster “strap-on” stages 
attached to the bottom stage of a vertical configuration (they can be either solid 
or liquid propellants) and this allows an increase in vehicle performance. The 
piggy-back configuration concept on the right is used on the Space Shuttle. The 
two large solid rocket motor boosters are not shown. 

Stage Separation 

It usually takes a finite time for the termination of the lower stage propulsion 
system to go from full thrust to essentially zero thrust (typically 1 to 3 sec for 
large thrust values). In some multistage flight vehicles (with stage separation 
devices) there often was a further short delay (say 4 to 10 sec) to achieve a 
respectable separation distance between the upper and the lower stage, before 
the firing of the upper stage propulsion system was initiated. This was done in 
order to prevent blow-back of damaging hot flames onto the upper stage. Also 
the upper stage engine start-up was not instantaneous but required one or more 
seconds. During this cumulative delay of several seconds the earth’s gravity pull 
continues to diminish the vehicle’s upward velocity, causing a reduction of the 
flight velocity by perhaps 20 to 500 ft/sec (7 to 160 m/sec). A scheme called 
hot staging has been used to diminish this velocity loss and shorten the staging 
time interval. The upper stage propulsion system is actually started at low but 
increasing thrust before the lower stage propulsion system has been fully shut off 
or well before it reaches essentially zero thrust. There are special flame-resistant 
ducts in the interstage structure to allow the flame or hot exhaust gases of the 
upper stage engine to be safely discharged and deflected symmetrically prior to 
and immediately after the actual separation of the stages without harming the 
functional hardware of the vehicle. This hot staging scheme has been used on 
large multistage vehicles, such as the Titan II in the United States and certain 



*The three-stage Zenit space launch vehicle is assembled in the Ukraine with Russian rocket engines. 
It is transported to the United States, loaded on a special floating launch platform (Sea Launch 
Program), and launched by a team that is headed by the Boeing Company. 



4.7. FLIGHT VEHICLES 143 



launch vehicles in China and the Soviet Union, because it does improve the flight 
performance. 

For multistage vehicles the stage mass ratios, thrust levels, propulsion dura- 
tions, and location or travel of the center of gravity of the stages are usually 
optimized, often using a complex trajectory computer program. The high specific 
impulse rocket engine (e.g., using hydrogen-oxygen propellants) is normally 
employed in upper stages of space launch vehicles, because a small increase in 
specific impulse can be more effective there than in lower stages. 

Example 4-3. A two-stage exploration vehicle is launched from a high-orbit satellite 
into a gravity-free vacuum trajectory. The following notation is used and explained in the 
accompanying diagram as well as in Fig. 4-1: 

mo = initial mass of vehicle (or stage) at launch 
mp = useful propellant mass of stage 
nii = initial mass of stage(s) 

nif = final mass after rocket operation; it includes the empty propulsion system with its 
residual propellant, the vehicle structures plus the propulsion system with control, 
guidance and payload masses 

mpi = payload mass; it can include guidance, control and communications equipment, 
antennas, scientific instruments, military equipment, research apparatus, power 
supply, solar panels, sensors, etc. 




144 FLIGHT PERFORMANCE 



Subscripts 1 and 2 refer to the first and second stages. The following data are given: 

Flight velocity increment in gravity-free vacuum 4700 m/sec 
Specific impulse (each stage) 310 sec 

Initial takeoff launch vehicle mass 4500 kg 

Propellant mass fraction, t; (each stage) 0.88 



Determine the payload for the following cases: 

1. When the two propulsion system or stage masses are equal [(m,)i = (WiOa]- 

2. When the mass ratios of the two stages are equal [(my/mo)i = [nif /mo) 2 ]. 

SOLUTION. The following relationships apply to both cases. The takeoff mass or launch 
mass can be divided into three parts, namely, the two propulsion stages and then payload 



(mo) = (m;)i + (m,)2 + (?«)pl 

The propellants are 88 % of their propulsion system mass, and this is representative of 
a number of stages. 



(mp)i = 0 . 88 (m;)i and (m ^)2 = 0 . 88 (m ,)2 

The nozzle exit area ratio and the chamber pressure are the same in both stages and 
both cases. Thus the exhaust velocities are the same. 

Cl = C2 = c = Isgo = 310 X 9.807 = 3040 m/sec 

Case 1. The stage masses and sizes are the same, or 

(m,)i = (m,)2 = nii 



Equation 4-36 can be rewritten 

gA„/c ^ (i/iv«j)(i/iv«)2 = {[m„ - (mp)i]/m„)[(m.)2 + mpi - (mpi)] 

gS500/3040 ^ [( 45 QQ _ o.88mi)/4500][m; - mpi - 0.88mi]/(m; -f mpi) = 6.105 
From the equation above m, = i(4500 — mpi). 

These two equations have two unknowns: m; and mpi. The solution requires a quadratic 
equation, and the value of the payload can be determined as mpi = 480 kg. 

Case 2. The mass ratios for the two stages are equal, or 



IVRi = ]VR2 = IVR = nto/nif 

1/NR = [4500 - (mp)i]/4500 = [(m ;)2 + mpi - (mp) 2 ]/[(mj )2 -H mpi] 



,Am/c _ 



4500 — 0.88(m,)i 0.12(m,)2 -f mpi 
4500 (m;)2 + mpi 



= ^ 5500/3040 ^ 



4500 



4.7. FLIGHT VEHICLES 145 



These two equations (and the first equation of the example makes a third) have three 
variables: (m,) 2 , and m^\. The solution is possible and requires quadratic equations 

and natural logarithms. The result is a payload of approximately 600 kg. 

If a three-stage vehicle had been used in Example 4-3 instead of a two- 
stage version, the payload would have been even larger. However, the theoretical 
payload increase will only be about 8 or 10%. A fourth stage gives an even 
smaller theoretical improvement; it would add only 3 to 5% to the payload. The 
amount of potential performance improvement diminishes with each added stage. 
Each additional stage means extra complications in an actual vehicle (such as a 
reliable separation mechanism, an interstage structure, more propulsion systems, 
joints or couplings in connecting pipes and cables, etc.), requires additional inert 
mass (increasing the mass ratio IVR), and compromises the overall reliability. 
Therefore, the minimum number of stages that will meet the payload and the Am 
requirements is usually selected. 

The flight paths taken by the vehicles in the two simplified cases of 
Example 4-3 are different, since the time of flight and the acceleration histories 
are different. One conclusion from this example applies to all multistage 
rocket-propelled vehicles; for each mission there is an optimum number of 
stages, an optimum distribution of the mass between the stages, and usually also 
an optimum flight path for each design, where a key vehicle parameter such as 
payload, velocity increment, or range is a maximum. 

Launch Vehicles 

Usually first or lowest stage, often called a booster stage, is the largest and it 
requires the largest thrust and largest total impulse. Eor earth surface launch all 
stages now use chemical propulsion to achieve the desired thrust-to-weight ratio. 
These thrusts usually become smaller with each subsequent stage, also known 
as upper stage or sustainer stage . The thrust magnitudes depend on the mass of 
the vehicle, which in turn depends on the mass of the payload and the mission. 
Typical actual configurations are shown by simple sketches in Fig. 4-14. There 
is an optimum size and thrust value for each stage in a multistage vehicle and 
the analysis to determine these optima can be quite complex. 

Many launch vehicles with heavy payloads have one to six large strap-on 
stages, also called zero stages or half stages. They augment the thrust of the 
booster stage, which is started at about the same time. A schematic diagram 
is shown in the third sketch of Fig. 4-14. Solid propellant strap-on stages are 
common, such as the Atlas V shown in Fig. 1-13 or the Space Shuttle shown in 
Fig. 1-14. They are usually smaller in size than the equivalent liquid propellant 
strap-on (due to higher propellant density) and have less drag and usually a very 
toxic exhaust. Liquid propellant strap-on stages are used in the Delta IV heavy 
lift launch vehicle (see Fig. 1-12), in the first Soviet ICBM (intercontinental 
ballistic missile, 1950s), and several Soviet/Russian space launch vehicles. They 



146 FLIGHT PERFORMANCE 



deliver higher specific impulse, which enhances vehicle performance, and require 
propellant filling at the launch site. 

There is a variety of existing launch vehicles. The smaller ones are for low 
payloads and low orbits; the larger ones usually have more stages, are heav- 
ier and more expensive, and have larger payloads or higher mission velocities. 
The vehicle cost increases with the number of stages and the initial vehicle 
launch mass. Once a particular launch vehicle has been proven to be reliable, 
it is usually modified and uprated to allow improvements in its capability or 
mission flexibility. Each of the stages of a space launch vehicle can have sev- 
eral rocket engines, each for specific missions or maneuvers. The Space Shuttle 
system has 67 different rockets which are shown schematically in Fig. 1-14. 
In most cases each rocket engine is used for a specific maneuver, but in many 
cases the same engine is used for more than one specific purpose; the small reac- 
tion control thrusters in the Shuttle serve, for example, to give attitude control 
(pitch, yaw, and roll) during orbit insertion and reentry, for counteracting internal 
shifting of masses (astronaut movement, extendible arm), small trajectory correc- 
tions, minor flight path adjustments, docking, and precise pointing of scientific 
instruments. 

The spacecraft is that part of a launch vehicle that carries the payload. It is the 
only part of the vehiele that goes into orbit or deep space and some are designed 
to return to earth. The final major space maneuver, such as orbit injection or 
planetary landing, often requires a substantial velocity increment; the propulsion 
system, which provides the force for this maneuver, may be integrated with the 
spacecraft or it may be part of a discardable stage, just below the spacecraft. 
Several of the maneuvers described in Section 4.5 can often be accomplished by 
propulsion systems located in two different stages of a multistage vehicle. The 
selection of the most desirable propulsion systems, and the decision of which 
of the several propulsion systems will perform specific maneuvers, will depend 
on optimizing performance, cost, reliability, schedule, and mission flexibility as 
described in Chapter 19. 

When a space vehicle is launched from the earth’s surface into an orbit, it flies 
through three distinct trajectory phases. (1) Most are usually launched vertically 
and then undergo a turning maneuver while under rocket power to point the flight 
velocity vector into the desired direction. (2) The vehicle then follows a free- 
flight (unpowered) ballistic trajectory (usually elliptical), up to its apex. Finally 
(3) a satellite needs an extra push from a chemical rocket system to add enough 
total impulse or energy to accelerate it to orbital velocity. This last maneuver 
is also known as orbit insertion or sometimes as a kick maneuver. During the 
initial powered flight the trajectory angle and the thrust cutoff velocity of the last 
stage are adjusted by the guidance system to a velocity vector in space that will 
allow the vehicle to reach the apogee of its elliptic path exactly at the desired 
orbit altitude. As shown in Fig. 4-9, a multistage ballistic missile follows the 
same two ascent flight phases mentioned above, but it then continues its elliptical 
ballistic trajectory all the way down to the target. 



4.7. FLIGHT VEHICLES 147 



500 



450 



400 



g 350 

o 

o 

~ 300 

a> 

o 

« 

Q- 



250 



200 



150 



100 





^0 degrees inclination, equ 

1 


atorial drop pt 


)int 






^28.5 degree 
s^^^38 de 


S(ER) 

‘grees (WFF) 












/ 70 degree 

/ 

\ /S 


s (WR) 

)0 degrees (W 
/Sun-! 


R), polar orbi 
Synchronous 










WR) 














Assumes: 

- 220 ft/sec 


Velocity Reserv 










System Kept on Launch V 

- Direct Injection (No Dog-L 
ER = Eastern Range 

- WEE = Wallops Elight Eac 

- WR = Western Range (Va 

^ 


3hicle Side 
3gs) 








lity 

ndenberg Air Fc 


rce Base) 

1 







200 400 600 800 1000 

Circular orbit altitude (km) 



1200 



1400 



FIGURE 4-15. Decrease of payload with circular orbit altitude and orbit inclination for 
early version of the Pegasus launch vehicle. This is an air-launched, relatively simple, 
three-stage launch vehicle of 50 in. diameter driven by a solid propellant rocket motor in 
each stage. (Courtesy Orbital Sciences Corporation.) 



Historically successful launch vehicles have been modified, enlarged, and 
improved in performance. The newer versions retain most of the old, proven, reli- 
able components, materials, and subsystems. This reduces development effort and 
cost. Upgrading a vehicle allows an increase in mission energy (more ambitious 
mission) or payload or both. Typically, it is done by one or more of these types 
of improvement: increasing the mass of propellant without an undue increase in 
tank or case mass; uprating the thrust and strengthening the engine; more specific 
impulse; or adding successively more or bigger strap-on boosters. It also usually 
includes a strengthening of the structure to accept higher loads. 

Figure 4-15 shows the effects of orbit inclination and altitude on the payload. 
The inclination is the angle between the equatorial plane of the earth and the 
trajectory. An equatorial orbit has zero inclination and a polar orbit has 90° 
inclination. Since the earth’s rotation gives the vehicle an initial velocity, a launch 
from the equator in an eastward direction will give the highest payload. For the 
same orbit altitude other trajectory inclinations have a lower payload. For the 
same inclination the payload decreases with orbit altitude, since more energy has 
to be expended to overcome gravitational attraction. 



148 FLIGHT PERFORMANCE 



The Space Shuttle has its maximum payload when launched due east into 
an orbit with 28.5° inclination from Kennedy Space Flight Center in Florida, 
namely about 56,000 lb (or 25,455 kg) at a 100-nautical-mile (185-km) orbit 
altitude. The payload decreases by about 100 lb (45.4 kg) for every nautical mile 
increase in altitude. If the inclination is 57°, the payload diminishes to about 
42,000 lb (or 19,090 kg). If launched in a southerly direction from Vandenberg 
Air Force Base on the west coast in a 98° inclination into a circular, nearly polar 
orbit, the payload will be only about 30,600 lb or 13,909 kg. 

The dramatic decrease of payload with circular orbits of increasing altitude 
and with different inclination is shown for the Pegasus, a relatively small, air- 
launched space launch vehicle, in Fig. 4-15. The payload is a maximum when 
launching from the earth equator in the east direction, that is, at 0° inclination. 
The figure shows that a practical payload becomes too small for orbits higher 
than about 1200 km. To lift heavier payloads and to go to higher orbits requires a 
larger launch vehicle than this Pegasus. Figure 4-15 is based on the assumption 
of a particular payload separation mechanism (38 in.) and a specific Am vehicle 
velocity reserve (220 ft/sec) for items such as the normal changes in atmospheric 
density (which can double the drag) or mass tolerances of the propulsion systems. 
Similar curves can usually be provided by the makers of all launch vehicles. 



4.8. MILITARY MISSILES 

The majority of all rocket propulsion systems built today are for military pur- 
poses. There is a large variety of missiles and military missions and therefore 
many different propulsion systems. All are chemical propulsion systems. They 
range from simple, small, unguided, fin-stabilized single-stage rocket projectiles 
(used in air-to-surface missions and surface-to-surface bombardment) up to com- 
plex, sophisticated, expensive, long-range, multistage ballistic missiles, which 
are intended for faraway military or strategic targets. The term “surface” means 
land surface (ground launch or ground target), ocean surface (ship launched), 
or below the ocean surface (submarine launched). A tactical missile is used for 
attacking or defending ground troops, nearby military or strategic installations, 
military aircraft, or war missiles. The armed forces also use military satellites 
for missions such as reconnaissance, early warning of impending attack, secure 
communication, or navigation. 

Strategic missiles with a range of 3000 km or more have been two- or 
three-stage surface-to-surface rocket-propelled missiles. Early designs used liq- 
uid propellant rocket engines and some are still in service in certain countries. 
Beginning about 37 years ago, newer strategic missiles have used solid propellant 
rocket motors in the United States and France. Both types usually also have a 
liquid propellant RCS for accurately adjusting the final payload flight velocity 
(in magnitude, direction, and position in space) at the cutoff of the propulsion 
system of the last stage. A solid propellant RCS version also exists. The flight 
analysis and ballistic trajectories of the long-range missiles are similar in many 
ways to those described for launch vehicles in this chapter. 



4.8. MILITARY MISSILES 149 



Solid propellant rocket motors are preferred for most tactical missile missions, 
because they allow simple logistics and can be launched quickly. If altitudes are 
low and flight durations are long, such as with a cruise missile, an air-breathing jet 
engine and a winged vehicle, which provides lift, will usually be more effective 
than a long-duration rocket. However, a large solid propellant rocket motor may 
still he needed as a booster to launch the cruise missile and bring it up to speed. 

For each of the tactical missile applications, there is an optimum rocket propul- 
sion system and almost all of them use solid propellant rocket motors. Liquid 
propellant rocket engines have recently been used for the upper stages of two- 
stage anti-aircraft missiles and ballistic defense missiles, because they can he 
pulsed for different durations and can he randomly throttled. For each appli- 
cation there is an optimum total impulse, an optimum thrust time profile, an 
optimum nozzle configuration (single or multiple nozzles, with or without thrust 
vector control, optimum area ratio), optimum chamber pressure, and a favored 
solid propellant grain configuration. Low exhaust plume gas radiation emissions 
in the visible, infrared, or ultraviolet spectrum and certain safety features (making 
the system insensitive to energy stimuli) can be very important in some of the 
tactical missile applications; these are discussed in Chapters 13 and 20. 

Short-range, uncontrolled, unguided, single-stage rocket vehicles, such as mil- 
itary rocket projectiles (ground and air launched) and rescue rockets, are usu- 
ally quite simple in design. Their general equations of motion are derived in 
Section 4.3, and a detailed analysis is given in Ref. 4-1. 

Unguided military rocket-propelled missiles are today produced in larger num- 
bers than any other category of rocket-propelled vehicles. The 2. 75-in. -diameter, 
folding fin unguided solid propellant rocket missile has recently been produced 
in the United States in quantities of almost 250,000 per year. Guided missiles for 
anti-aircraft, antitank, or infantry support have been produced in annual quanti- 
ties of hundreds and sometimes over a thousand. Table 1-6 lists several guided 
missiles. 

Because these rocket projectiles are essentially unguided missiles, the accuracy 
of hitting a target depends on the initial aiming and the dispersion induced by 
uneven drag, wind forces, oscillations, and misalignment of nozzles, body, and 
fins. Deviations from the intended trajectory are amplified if the projectile is 
moving at a low initial velocity, because the aerodynamic stability of a projectile 
with fins is small at low flight speeds. When projectiles are launched from an 
aircraft at a relatively high initial velocity, or when projectiles are given stability 
by spinning them on their axis, their accuracy of reaching a target is increased 
2- to 10-fold, compared to a simple fin-stabilized rocket launched from rest. 

In guided air-to-air and surface-to-air rocket-propelled missiles the time of 
flight to a given target, usually called the time to target f,, is an important flight 
performance parameter. With the aid of Fig. 4- 16 it can be derived in a simplified 
form by considering the distance traversed by the rocket (called the range) to 
be the integrated area underneath the velocity-time curve. This simplification 
assumes no drag, no gravity effect, horizontal flight, a relatively small distance 
traversed during powered flight compared to the total range, and a linear increase 



150 FLIGHT PERFORMANCE 





Powered 
















ll 








>. 


j ! 


Probable actual curve ' 






o 


/' 


allowing for drag and 


p 




d> 




non-linear change in 








/ 


mass 












I 








1 



tp Time f, 



Maximum velocity 
at burnout of 
propellant 

Actual velocity is 
decreased by 
drag 

Velocity of 

launching 

aircraft 



FIGURE 4-16. Simplified trajectory for an unguided, nonmaneuvering, air-launched 
rocket projectile. Solid fine shows ideal flight velocity without drag or gravity and dashed 
curve shows likely actual flight. 



in velocity during powered flight: 



S' -j- 2^p^p 
Mo + Up 



(4-37) 



Here S is the flight vehicle’s range to the target and it is the integrated area 
under the velocity -time curve. Also Up is the velocity increase of the rocket 
during powered flight up to the time of burnout, tp is the time of rocket burning, 
and uq is the initial velocity of the launching aircraft. For the same flight time the 
range of the actual vehicle (dashed line) is less than for the ideal dragless vehicle. 
For more accurate values, the velocity increase Up is given by Eq. 4-19. More 
accurate values can also be obtained through a detailed step-by-step trajectory 
analysis that considers the effects of drag and gravity. 

In unguided air-to-air or air-to-surface rocket-powered projectiles the aiming 
at the target is done largely by orienting the launching aircraft into the direction of 
the target. A relatively simple solid propellant rocket motor is the most common 
choice for the propulsion. In guided missiles, such as air-to-air, air-to-ground, or 
ground-to-air, the flight path to the target has to be controlled and this can be 
achieved by controlling aerodynamic control surfaces and/or propulsion systems, 
which can be pulsed (repeated start and stop) and/or throttled to a lower thrust. 
The guidance system and the target seeker system of a guided missile will sense 
and track the flight path of a flying target, a computer will calculate a predicted 
impact point, and the missile’s flight control will change the flight path of the 
guided missile to achieve the impact with the intended target. The control system 
will command the propulsion system to operate or fire selected liquid propellant 
thrusters of an engine with multiple thrusters (or to selectively provide thrust 
through multiple nozzles with hot-gas shut-off valves in solid motors). A sim- 
ilar set of events can occur in a defensive ground-to-incoming-ballistic-missile 



4.8. MILITARY MISSILES 151 



scenario. It requires propulsion systems capable of pulsing or repeated starts, pos- 
sibly with some throttling and side forces. Rocket engines with these capabilities 
are discussed in Section 6.8. 

In both the unguided projectile and the guided missile the hit probability 
increases as the time to target t, is reduced. In one particular air-to-air combat 
situation, the effectiveness of the rocket projectile varied approximately inversely 
as the cube of the time to target. The best results (e.g., best hit probability) are 
usually achieved when the time to target is as small as practically possible. 

The analysis of the missile and propulsion configuration that gives the min- 
imum time to target over all the likely flight scenarios can be complex. The 
following rocket propulsion features and parameters will help to reduce the time 
to target, but their effectiveness will depend on the specific mission, range, guid- 
ance and control system, and particular flight conditions. 

1. High initial thrust or high initial acceleration for the missile to quickly 
reach a high-initial-powered flight velocity. 

2. Application of additional lower thrust to counteract drag and gravity losses 
and thus maintain a high flight velocity. This can be a single rocket propul- 
sion system that has a short high initial thrust and a smaller (10 to 25%) 
sustaining thrust of longer duration. 

3. For higher supersonic flight speeds, a two-stage missile can be more effec- 
tive. Here the first stage is dropped off after its propellant has been con- 
sumed, thus reducing the inert mass of the next stage and improving its 
mass ratio and thus its flight velocity increase. 

4. If the target is highly maneuverable and if the closing velocity between 
missile and target is large, it may be necessary not only to provide an 
axial thrust but also to apply large side forces or side accelerations to a 
defensive tactical missile. This can be accomplished either by aerodynamic 
forces (lifting surfaces or flying at an angle of attack) or by multiple-nozzle 
propulsion systems with variable or pulsing thrusts; the rocket engine then 
would have an axial thruster and one or more side thrusters. The side 
thrusters have to be so located that all the thrust forces are essentially 
directed through the center of gravity of the vehicle in order to minimize 
turning moments. The thrusters that provide the side accelerations have 
also been called divert thrusters, since they divert the vehicle in a direction 
normal to the axis of the vehicle. 

5. Drag losses can be reduced if the missile has a large L/D ratio (or a small 
cross-sectional area) and if the propellant density is high, allowing a smaller 
missile volume. The drag forces can be high if the missile travels at low 
altitude and high speed. A long and thin propulsion system geometry and 
a high-density propellant will help to reduce drag. 

A unique military application is rocket-assisted gun-launched projectiles for 
attaining longer artillery ranges. Their small rocket motors withstand very high 



152 FLIGHT PERFORMANCE 



accelerations in the gun barrel (5000 to 10,000 go is typical). They have been in 
production. 



4.9. FLIGHT STABILITY 

Stability of a vehicle is achieved when the vehicle does not rotate or oscil- 
late in flight. Unstable flights are undesirable, because pitch or yaw oscillations 
increase drag (flying at an angle of attack most of the time) and cause prob- 
lems with instruments and sensors (target seekers, horizon scanners, sun sensors, 
or radar). Instability often leads to tumbling (uncontrolled turning) of vehicles, 
which causes missing of orbit insertion, missing targets, or sloshing of liquid 
propellant in tanks. 

Stability can be built in by proper design so that the flying vehicle will be 
inherently stable, or stability can be obtained by appropriate controls, such as 
the aerodynamic control surfaces on an airplane, a reaction control system, or 
hinged multiple rocket nozzles. 

Flight stability exists when the overturning moments (e.g., those due to a wind 
gust, thrust misalignment, or wing misalignment) are smaller than the stabilizing 
moments induced by thrust vector controls or by aerodynamic control surfaces. 
When the destabilizing moments exceed the stabilizing moments about the center 
of gravity, the vehicle turns or tumbles. In unguided vehicles, such as low-altitude 
rocket projectiles, stability of flight in a rectilinear motion is achieved by giving 
a large stability margin to the vehicle by using tail fins and by locating the center 
of gravity ahead of the center of aerodynamic pressure. In a vehicle with an 
active stability control system, a nearly neutral inherent stability is desired, so 
that the applied control forces are small, thus requiring small control devices, 
small RCS thrusters, small actuating mechanisms, and structural mass. Neutral 
stability is achieved by locating aerodynamic surfaces and the mass distribution 
of the components within the vehicle in such a manner that the center of gravity is 
only slightly above the center of aerodynamic pressure. Because the aerodynamic 
moments change with Mach number, the center of pressure does not necessarily 
stay fixed during accelerating flight but shifts, usually along the vehicle axis. 
The center of gravity also changes its position as propellant is consumed and the 
vehicle mass decreases. Thus it is usually very difficult to achieve neutral missile 
stability at all altitudes, speeds, and flight conditions. 

Stability considerations affect rocket propulsion system design in several ways. 
By careful nozzle design and careful installation it is possible to minimize thrust 
misalignment and thus to minimize undesirable torques on the vehicle and the 
reaction control propellant consumption. It is possible to exercise considerable 
control over the travel of the center of gravity by judicious design. In liquid 
propellant rockets, special design provisions, special tank shapes, and a careful 
selection of tank location in the vehicle afford this possibility. By using nozzles 
at the end of a blast tube, as shown in Fig. 15-6, it is possible to place the solid 
propellant mass close to the vehicle’s center of gravity. Attitude control liquid 



PROBLEMS 153 



propellant engines with multiple thrusters have been used satisfactorily to obtain 
control moments for turning vehicles in several ways, as described in Section 4.5 
and in Chapter 6. 

Unguided rocket projectiles and missiles are often given a roll or rotation 
by inclined aerodynamic fins or inclined multiple rocket exhaust gas nozzles 
to improve flight stability and accuracy. This is similar to the rotation given 
to bullets by spiral-grooved barrels. This spin stability is achieved in part by 
gyroscopic effects, where an inclination of the spin axis is resisted by torques. The 
centrifugal effects cause problems in emptying liquid propellant tanks and extra 
stresses on solid propellant grains. In some applications a low-speed roll is applied 
not for spin stability but to assure that any effects of thrust vector deviations or 
aerodynamic vehicle shape misalignments are minimized and canceled out. 



PROBLEMS 

1. For a vehicle in gravitationless space, determine the mass ratio necessary to boost the 
vehicle velocity by (a) 1600 m/sec and (b) 3400 m/sec; the effective exhaust velocity 
is 2000 m/sec. If the initial total vehicle mass is 4000 kg, what are the corresponding 
propellant masses? 

Answers: (a) 2204 kg. 

2. Determine the burnout velocity and burnout altitude for a dragless projectile with the 
following parameters for a simplified vertical trajectory: c = 2209 m/sec; nip/mo = 
0.57; tp = 5.0 sec; and mq = 0; Hq = 0. Select a relatively small diameter missile with 
UD of 10 and an average vehicle density of 1200 kg/m^. 

3. Assume that this projectile had a drag coefficient essentially similar to the 0° curve in 
Fig. 4-3 and redetermine the answers of Problem 3 and the approximate percentage 
errors in Up and hp. Use a step-by-step or a numerical method. 

4. A research space vehicle in gravity-free and drag-free outer space launches a smaller 
spacecraft into a meteor shower region. The 2-kg sensitive instrument package of 
this spacecraft (25 kg total mass) limits the maximum acceleration to no more than 
50 m/sec^. It is launched by a solid propellant rocket motor (/j = 260 sec and f = 
0.88). Assume instant start and stop of rocket motor. 

(a) Determine the maximum allowable burn time, assuming steady constant propel- 
lant mass flow; 

(b) Determine the maximum velocity relative to the launch vehicle. 

(c) Solve for (a) and (b) if half of the total impulse is delivered at the previous 
propellant mass flow rate, with the other half at 20% of this mass flow rate. 

5. For a satellite cruising in a circular orbit at an altitude of 500 km, determine the 
period of revolution, the flight speed, and the energy expended to bring a unit mass 
into this orbit. 

Answers: 1.58 hr, 7613 m/sec, 33.5 MJ/kg. 

6. A large ballistic rocket vehicle has the following characteristics: propellant mass flow 
rate: 12 slugs/sec (1 slug = 32.2 Ibm = 14.6 kg); nozzle exit velocity: 7100 ft/sec; 



154 FLIGHT PERFORMANCE 



nozzle exit pressure: 5 psia (assume no separation); atmospheric pressure: 14.7 psia 
(sea level); takeoff weight; 12.0 tons (Iton = 2000 Ihf); burning time; 50 sec; nozzle 
exit area: 400 in.^ Determine (a) the sea-level thrust; (b) the sea-level effective 
exhaust velocity; (c) the initial thrust-to-weight ratio; (d) the initial acceleration; (e) 
the mass inverse ratio mo/ntf. 

Answers: 81,320 Ibf; 6775 ft/sec; 3.38; 2.38go- 

7. In Problem 5 compute the altitude and missile velocity at the time of power plant 
cutoff, neglecting the drag of the atmosphere and assuming a simple vertical 
trajectory. 

8. A spherical satellite has 12 identical monopropellant thrust chambers for attitude 
control with the following performance characteristics: thrust (each unit): 5 Ibf; 4 
(steady state or more than 2 sec): 240 sec; 4 (pulsing duration 20 msec): 150 sec; 
4 (pulsing duration 100 msec): 200 sec; satellite weight: 3500 Ibf; satellite diameter: 
8 ft; satellite internal density distribution is essentially uniform; disturbing torques, 
Y and Z axes: 0.00005 ft-lbf average; disturbing torque, for X axis: 0.001 ft-lbf 
average; distance between thrust chamber axes: 8 ft; maximum allowable satellite 
pointing position error: ±1°. Time interval between pulses is 0.030 sec. 

(a) What would be the maximum and minimum vehicle angular drift per hour if no 
correction torque were applied? 

Answers: 0.466 and 0.093 rad/hr. 

(b) What is the frequency of pulsing action (how often does an engine pair operate?) 
at 20-msec, 100-msec, and 2-sec pulses in order to correct for angular drift? 
Discuss which pulsing mode is best and which is impractical. 

9 . For an ideal multistage launch vehicle with several stages in series, discuss the fol- 
lowing: (a) the effect on the ideal mission velocity if the second and third stages are 
not started immediately but are each allowed to coast for a short period after shutoff 
and separation of the expended stage before rocket engine start of the next stage; 
(b) the effect on the mission velocity if an engine malfunctions and delivers a few 
percent less than the intended thrust but for a longer duration and essentially the full 
total impulse of that stage. 

10 . Given a cylindrically shaped space vehicle (D = Im, height is 0.7 m, average density 
is 1.1 g/cm^) with a flat solar cell panel on an arm (mass of 32 kg, effective moment 
arm is 1.5 m, effective average area facing normally toward sun is 0.6 m^) in a set 
of essentially frictionless bearings and in a low circular orbit at 160 km altitude with 
sunlight being received, on the average, about 54% of the time. The top surface of 
the cylinder and the surface of the solar have solar cells. The satellite is maneuvered 
so that these two surfaces are always perpendicular to the sun’s rays. The articulated 
arm between the cylinder and the solar panel is small and its mass and force, due to 
solar radiation pressure, are also small and can be neglected. 

(a) Compute the maximum solar pressure-caused torque and the angular displacement 
this would cause during 1 day if not corrected. 

(b) Using the data from the atmospheric table in Appendix 2 and an arbitrary average 
drag coefficient of 1.0 for both the body and the flat plate, compute the drag force 
and torque. 



SYMBOLS 155 



(c) Using stored high-pressure air at 14 x 10® N/m^ initial pressure as the propellant 
for attitude control, design an attitude control system to periodically correct for 
these two disturbances etc.). 

11 . Determine the payload for a single-stage vehicle in Example 4-3. Using the data 
from this example compare it with the two-stage vehicle. 

12 . An earth satellite is in an elliptical orbit with the perigee at 600 km altitude and an 
eccentricity of e = 0.866. Determine the parameters of the new satellite trajectory, 
if a rocket propulsion system is fired in the direction of flight giving an incremental 
velocity of 200 m/sec when (a) fired at apogee, (b) fired at perigee, and (c) fired at 
perigee, but in the opposite direction, reducing the velocity. 

13 . A sounding rocket (75 kg mass, 0.25 m diameter) is speeding vertically upward at an 
altitude of 5000 m and a velocity of 700 m/sec. What is the deceleration in multiples 
of g due to gravity and drag? (Use Cd from Fig. 4-3 and use Appendix 2.) 

14 . Derive Eq. 4-37; state all your assumptions. 



SYMBOLS 

a 

A 

b 

B 

c 

c 

Cd 

Cl 

d 

D 

e 

e 

E 

F 

Ff 

F, 

Fo 

g 

go 

g 

G 

h 

hp 

h 

kii 

kg 

I 



major axis of ellipse, m, or acceleration, m/sec^ (ft/sec^) 

area, m^(ft^) 

minor axis of ellipse, m 

numerical value of drag integral 

effective exhaust velocity, m/sec (ft/sec) 

average effective exhaust velocity, m/sec 

drag coefficient 

lift coefficient 

total derivative 

drag force, N (Ibf) 

eccentricity of ellipse, e = y/\ — 

base of natural logarithm (2.71828) 

energy, J 

thrust force, N (Ibf) 
final thrust, N 

gravitational attraction force, N 

initial thrust force, N 

gravitational acceration, m/sec^ 

gravitational acceleration at sea level, 9.8066 m/sec^ 

average gravitational attraction, m/sec^ 

universal or Newton’s gravity constant, 

6.6700 X 10^' m^/kg-sec^ 
altitude, m (ft) 

altitude of rocket at power cutoff, m 
specific impulse, sec 
diffuse coefficient of reflectivity 
specular coefficient of reflectivity 
distance of moment arm, m 



156 FLIGHT PERFORMANCE 



L 

m 

nif 

Mp 

m 

m 

Ma 

Wi 

n 

P 

r 

R 

Ro 

S 

1 



It 

T 

u 

Un 



UQ 

Ve 

W 

X, y 



lift force, N (Ibf) 

instantaneous vehicle mass, kg (Ibm) 
final vehicle mass after rocket operation, kg 
useful propellant mass, kg 

initial vehicle launching mass, prior to rocket operation, kg 

mass flow rate of propellant, kg/sec 

angular moment of inertia, kg-m^ 

mass ratio of vehicle = trif /mo 

number of stages 

pressure, N/m^ or Pa (psi) 

radius, m, or distance between the centers of two attracting 
masses, m 

instantaneous radius from vehicle to center of earth, m 
effective mean earth radius, 6.3742 x 10^ m 
range, m 
time, sec 

time from launching to power cutoff or time from propulsion 

start to thrust termination, sec 

time to target, sec 

torque, N-m (ft-lbf) 

vehicle flight velocity, m/sec (ft/sec) 

orbital velocity at apogee, m/sec 

velocity at power cutoff, m/sec, or orbital velocity at perigee, 
m/sec 

initial or launching velocity, m/sec 
escape velocity, m/sec 
weight, N (in some problems, Ibf) 
any points on an elliptical orbit 



Greek Letters 

a angle of attack, deg or rad, or angular acceleration, angle/sec^ 

i; propellant mass fraction (f = mp/mo) 

9 angle between flight direction and horizontal, or angle of incident 

radiation, deg or rad 

/X gravity constant for earth, 3.98600 x lO'^ m^/sec^ 

p mass density, kg/m^ 

T period of revolution of satellite, sec 

t]/ angle of thrust direction with horizontal 

CO angular speed, deg/sec (rad/sec) 



Subscripts 



e 

f 

i 



escape condition 

final condition at rocket thrust termination 
initial condition 



REFERENCES 157 



max 

P 



maximum 

condition at power cutoff or propulsion termination 



pi 



payload 

satellite 

zenith 



z 



0 



initial condition or takeoff condition 



REFERENCES 

4-1. J. B. Rosser, R. R. Newton, and G. L. Gross, Mathematical Theory of Rocket 
Flight, McGraw-Hill Book Company, New York, 1947; or F. R. Gantmakher and 
L. M. Levin, The Flight of Uncontrolled Rockets, Macmillan, New York, 1964. 

4-2. R. S. Wolf, “Development of a Handbook for Astrobee F (Sounding rocket) Flight 
Performance Predictions,” Journal of Spacecraft and Rockets, Vol. 24, No. 1, 
January -February 1987, pp. 5-6. 

4-3. Orbital Flight Handbook, NASA SP33, 1963, Part 1: Basic Techniques and Data. 
Part 2: Mission Sequencing Problems. Part 3: Requirements. 

4-4. V. A. Chobotov (Ed) Orbital Mechanics , 3rd ed.. Educational Series, AIAA, 
Reston, VA., 2002; and T. Logsdon, Orbital Mechanics and Application, John 
Wiley & Sons, New York, October 1997. 

4-5. J. W. Cornelisse, H. F. R. Schoyer, and K. F. Wakker, Rocket Propulsion and 
Space Flight Dynamics , Pitman Publishing, Londn, 1979. 

4-6. J. P. Vinti, G. J. Der, and L. Bonavito, Orbits and Celestial Mechanics, Vol. 177 
of Progress in Aeronautics and Astronautics Series, AIAA, Reston, VA, 1998, 
409 pages. 

4-7. W. Hohmann, Die Erreichbarkeit der Himmelskbrper (Accessibility of Celestial 
Bodies), Oldenburg, Munich, 1925. 

4-8. W. J. Larson and J. R. Wertz, Space Mission Analysis and Design, 3rd ed., pub- 
lished jointly by Microcosm, Inc. and Kluwer Academic Press, 1999. 

4-9. 1. 1. Pocha, An Introduction to Mission Design for Geostationary Satellites, Kluwer 
Academic Publishers, Hingham, MA, 1987, 222 pages. 

4-10. M. H. Kaplan, Orbital Spacecraft Dynamics and Control, John Wiley & Sons, 
New York, 1976. 

4-11. R. W. Humble, G. N. Henry, and W. J. Larson, Space Propulsion Analysis and 
Design, McGraw-Hill, New York, 1995, 748 pages. 

4-12. J. P. Vinti, G. J. Der, and L. Bonavito, Orbit and Celestial Mechanics, Vol. 177 
of Progress in Astronautics and Aeronautics Series, AIAA, Reston, VA, 1998, 
409 pages. 

4-13. C-C. G. Chao, Applied Orbit Perturbations and Maintenance, Aerospace Press, 
2005, 264 pages. 

4-14. “Spacecraft Aerodynamic Torques,” NASA SP 8058, January 1971 (N 71-25935). 

4-15. “Spacecraft Radiation Torques,” NASA SP 8027, October 1969 (N 71-24312). 

4-16. “Spacecraft Gravitational Torques,” NASA SP 8024, May 1964 (N 70-23418). 

4-17. “Spacecraft Magnetic Torques,” NASA SP 8018, March 1969 (N 69-30339). 



CHAPTER 5 



CHEMICAL ROCKET PROPELLANT 
PERFORMANCE ANALYSIS 



In Chapter 3, simplified one-dimensional performance relations were developed. 
They require a knowledge of the composition of the hot rocket gas and 
the properties of the propellant reaction products, such as their combustion 
temperature T\, average molecular mass Tl, and the specific heat ratio or 
the enthalpy change {hi — /j 2 )- This chapter discusses several approaches to 
determine these theoretical thermochemical properties for a given composition 
or mixture of propellant, chamber pressure, nozzle shape, and nozzle exit 
pressure. This then allows the determination of performance parameters, such 
as theoretical specific impulse or exhaust velocity for chemical rockets. 

By knowing the calculated gas temperature, pressure, and gas composition 
it is possible to calculate other gas properties. This knowledge also allows 
the analysis and selection of materials for chamber and nozzle structures. 
Heat transfer analyses require the determination of the specific heats, thermal 
conductivity, and specific heat ratio for the gas mixture. The calculated exhaust 
gas composition forms the basis for estimating environmental effects, such as 
the potential spreading of a toxic cloud near a launch site, as discussed in 
Chapter 21. The exhaust gas parameters also form the basis for the analysis of 
exhaust plumes (Chapter 20) or flames external to the nozzle. 

With the advent of digital computers it has been possible to solve the set of 
equations involving mass balance, energy balance, together with thermodynamic 
and chemical equilibria of complex systems with a variety of propellant ingre- 
dients. This chapter is intended to introduce the background to this theoretical 
analysis, so the reader can understand the thermodynamic and chemical basis 
of the several computer programs that are in use today. This chapter does not 
describe any specific computer analysis programs. However, it discusses which 



158 



5.1. BACKGROUND AND FUNDAMENTALS 159 



of the physical phenomena or chemical reactions can or cannot be adequately 
simulated by computer analysis. 

The reader is referred to Refs. 5-1 to 5-5 for general chemical and thermo- 
dynamic background and principles. For a detailed description of the properties 
of each of the possible reactant and reaction products, see Refs. 5-6 to 5-12. 

All of these theoretical analyses are only approximations of what really 
happens in rocket combustion and nozzle flow, since they all require some 
simplifying assumptions. As more of the different phenomena are understood 
and mathematically simulated, the analysis approach and the computer imple- 
mentation become more realistic but also more complex. The 11 assumptions 
made in Section 3.1 for an ideal rocket are valid here also but only for a 
quasi-one-dimensional flows. However, more sophisticated analyses can make 
one or more of these assumptions unnecessary. The analysis is usually divided 
into two somewhat separate sets of calculations: 

1. The combustion process is the first part. It usually occurs in the combus- 
tion chamber at essentially constant chamber pressure (isobaric) and the 
resulting gases follow Dalton’s law, which is discussed in this chapter. 
The chemical reactions or the combustions occur very rapidly. The cham- 
ber volume is assumed to be large enough and the residence time in the 
chamber long enough for attaining chemical equilibrium in the chamber. 

2. The nozzle gas expansion process constitutes the second set of calculations. 
The fully reacted, equilibrated gas combustion products enter the nozzle 
and undergo an adiabatic expansion in the nozzle. The entropy remains 
constant during a reversible (isentropic) nozzle gas expansion, but in real 
nozzles it increases slightly. 

The principal chemical reactions occur inside the combustion chamber of a 
liquid propellant rocket engine or inside the grain cavity of a solid propellant 
rocket motor, usually within a short distance from the burning surface. These 
chamber combustion analyses are discussed further in Chapters 9 and 14. How- 
ever, some chemical reactions also occur in the nozzle as the gases expand; 
the composition of the reaction products can therefore change in the nozzle, as 
described in this chapter. A further set of chemical reactions can occur in the 
exhaust plume outside the nozzle, as described in Chapter 20; many of the same 
basic thermochemical analysis approaches described in this chapter also apply to 
exhaust plumes. 



5.1. BACKGROUND AND FUNDAMENTALS 

The analytical description of chemical reaction or combustion of one or more 
fuels with one or more oxidizing reactants is the basis of chemical rocket propul- 
sion. The heat liberated in this reaction transforms the propellants (reaetants) into 



160 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



hot gaseous products, which in turn are thermodynamically expanded in a nozzle 
to produce thrust. 

The chemical reactants or propellants can initially be either liquid or solid 
and occasionally also gaseous. The reaction products are usually gaseous, but 
with some propellants one or more reactant species remain in the solid or liquid 
phase. For example, with aluminized solid propellants, the chamber reaction gases 
contain liquid aluminum oxide and the colder gases in the nozzle exhaust contain 
solid, condensed aluminum oxide particles. For some of the chemical species, 
therefore, the analysis must consider as many as all three phases and the energy 
changes for the phase transitions must be included. When the amount of solid or 
liquid in the exhaust is small and the particles are small, to assume a perfect gas 
introduces only small errors. 

It is necessary to accurately know the chemical composition of the propel- 
lants and their relative proportion. In liquid propellant this means the mixture 
ratio and the major propellant impurities; in gelled or slurried liquid propellants 
it also includes suspended or dissolved solid materials; and in solid propellants 
it means all the ingredients, their proportions and impurities, and phase (some 
ingredients, such as plasticizers, can be in a liquid state). 

Dalton’s law applies to the gas resulting from the combustion. It states that 
a mixture of gases at equilibrium exerts a pressure that is the sum of the par- 
tial pressures of the individual gases, all at a common volume and a common 
temperature. The subscripts a,b,c, and so on refer to individual gas constituents: 



The perfect gas equation pV = RT applies very closely to high-temperature 
gases. Here V is the specific volume or the volume per unit mass of gas mixture, 
and the gas constant R for the mixture is obtained by dividing the universal 
gas constant R' (8314.3 J/kg-mol-K) by the average moleeular mass Wl (often 
erroneously called the molecular weight) of the gas mixture. Using Dalton’s law, 
Eq. 5-1 can be rewritten 



The volumetric proportions of gas species in a gas mixture are determined from 
the molar concentration or molar fractions , iij, expressed as kg-mol for a partic- 
ular species j per kg of mixture. If n is the total number of kg-mol of all species 
per kilogram of uniform gas mixture, then the mole fraction Xj 



P = Pa + Pb + Pc 

T = Ta = Tb = Tc=- 



(5-1) 

(5-2) 



p = RaTIVa + RbT/Vb + RcT/Vc + ■■■= R'TUmVr^^) (5-3) 



]=m 



nj 

Xj = — since 
n 




(5-4) 



5.1. BACKGROUND AND FUNDAMENTALS 161 



where nj is the kg-mol of species j per kilogram of mixture, m is the number 
of different gaseous species present in the equilibrium combustion gas products. 
The effective average molecular mass 971 of a gas mixture is then 



971 = 



E m 

j=l "7 



(5-5) 



There are n possible species which enter into the relationship and of these 
only m are gases, so n — m represents the number of condensed species. The 
molar specific heat for a gas mixture at constant pressure Cp can be determined 
from the individual gas molar fractions Uj and their molar specific heats as shown 
by Eq. 5-6. The specific heat ratio k of the mixture follows from Eq. 5-7: 



(6^»)mix — 






E m 

,/=i «;■ 
(6p)mix 



(Cp)mix - R' 



(5-6) 

(5-7) 



When a chemical reaction goes to completion, that is, all of the reactants are 
consumed and transformed into reaction products, the reactants are in stoichio- 
metric proportions. For example, consider this reaction: 



H2 + ^02 -> H 2 O (5-8) 

All the hydrogen and oxygen are fully consumed to form the single 
product — water vapor — without any reactant residue of either hydrogen or 
oxygen. In this ease it requires 1 mol of the H 2 and | mole of the O 2 to obtain 
1 mol of H 2 O. On a mass basis this stoichiometric mixture requires half of 
32.0 kg of O 2 and 2 kg of H 2 , which are in the stoichiometric mixture mass 
ratio of 8:1. The release of energy per unit mass of propellant mixture and the 
combustion temperature are highest at or near the stoichiometric mixture. 

Rocket propulsion systems usually do not operate with the proportion of their 
oxidizer and fuel in the stoichiometric mixture ratio. Instead, they usually operate 
fuel rich because this allows lightweight molecules such as hydrogen to remain 
unreacted; this reduces the average molecular mass of the reaction products, 
which in turn increases the specific impulse (see Eq. 3-16). For rockets using 
H 2 and O 2 propellants the best operating mixture mass ratio for high-performance 
rocket engines is typically between 4.5 and 6.0, not at the stoichiometric value 
of 8.0, because the drop in combustion temperature is small and there is more 
H 2 gas (low molecular mass) in the exhaust. 

Equation 5-8 is a reversible chemical reaction; by adding energy to the H 2 O 
the reaction can be made to go backward to create H 2 and O 2 and the arrow 
in the equation would be reversed. The decompositions of solid propellants into 
reaction product gases are irreversible chemical reactions, as is the reaction of 



162 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



liquid propellants burning to create gases. However, reactions among combustion 
product gases are usually reversible. 

Chemical equilibrium exists in reversible chemical reactions when the rate of 
forming products is exactly equal to the reverse reaction of forming reactants from 
the products. Once this equilibrium is reached, no further changes in concentration 
can take place. In Equation 5-8 all three gases would be present and their relative 
proportions would depend on the pressure, temperature, and initial mixture. 

The heat of formation AfH^ is the energy released (or absorbed), or the 
value of enthalpy change, when 1 mol of a chemical compound is formed from 
its constituent atoms or elements at 1 bar (100,000 Pa) and isothermally at 
298.15 K or 25°C. The A implies that it is an energy change. The subscript 
/ refers to formation and the superscript 0 means that each product or reactant 
substance is at its thermodynamic standard state and at the reference pressure and 
temperature. By convention, the heat of formation of the gaseous elements (e.g., 
H 2 , 02 , Ar, Xe, etc.) is set to zero at these standard conditions of temperature and 
pressure. Typical values of AjH^ and other properties are given in Table 5-1 
for selected species. When heat is absorbed in the formation of a product, then 
AfH^ has a positive value. Earlier analyses have been made with the standard 
temperature at other values, such as 273.15 K and a slightly higher standard 
reference pressure of 1 atm (101, 325 Pa). 

The heat of reaction ArH^ is the energy released or absorbed when products 
are formed from its reactants at standard reference conditions, namely at 1 bar and 
25°C. The heat of reaction can be negative or positive, depending on whether the 
reaction is exothermic or endothermic. The heat of reaction at other temperatures 
or pressures has to be corrected in accordance with the change in enthalpy. When 
a species changes from one state to another (e.g., liquid becomes gas or vice 
versa), it may lose or gain energy. In most rocket propulsion the heat of reaction 
is determined for a constant-pressure combustion process. In general the heat of 
reaction can be determined from sums of the heats of formation of the products 
and the reactants, namely 

ArH^ = ^ [Uj (A/H‘');]p„duct.s - Y. [”/■ (A/-f^‘’);]reactants (5-9) 

Here nj is the molar fraction of each particular species j. In a typical rocket 
propellant there are a number of different chemical reactions going on simul- 
taneously; Eq. 5-9 provides the heat of reaction for all of these simultaneous 
reactions. For data on heats of formation and heats of reaction, see Refs. 5-7 to 
5-13. 

Various thermodynamic criteria that represent the necessary and sufficient con- 
ditions for an equilibrium to be stable were first advanced by J. W. Gibbs early 
in the 20th century; they are based on minimizing the free energy. The Gibbs 
free energy G (often called the chemical potential) is a convenient derived func- 
tion or property of the state of a chemical material describing its thermodynamic 
potential and is directly related to the internal energy U , the pressure p, molar 
volume V, enthalpy h, temperature T, and entropy S. Eor a single species j the 



5.1. BACKGROUND AND FUNDAMENTALS 163 



TABLE 5-1. Chemical Thermodynamic Properties of Selected Substances at 298.15 K 
(25°C) and 0.1 MPa (1 bar) 



Substance 


Phase" 


Molar Mass 
(g/mol) 


AfH° 

(kj/mol) 


A/G® 

(kJ/mol) 


S° 

(J/mol-K) 


Cp 

(J/mol-K) 


A1 (crystal) 


s 


29.9815 


0 


0 


28.275 


24.204 


AI 2 O 3 


1 


101.9612 


-1620.567 


-1532.025 


67.298 


79.015 


C (graphite) 


s 


12.011 


0 


0 


5.740 


8.517 


CH 4 


g 


16.0476 


-74.873 


-50.768 


186.251 


35.639 


CO 


g 


28.0106 


-110.527 


-137.163 


197.653 


29.142 


CO 2 


g 


44.010 


-393.522 


-394.389 


213.795 


37.129 


H 2 


g 


2.01583 


0 


0 


130.680 


28.836 


HCl 


g 


36.4610 


-92.312 


-95.300 


186.901 


29.136 


HF 


g 


20.0063 


-272.546 


-274.646 


172.780 


29.138 


H 2 O 


1 


18.01528 


-285.830 


-237.141 


69.950 


75.351 


H 2 O 


g 


18.01528 


-241.826 


-228.582 


188.834 


33.590 


N 2 H 4 


1 


32.0451 


-H50.626 


149.440 


121.544 


98.840 


N 2 H 4 


g 


32.0451 


-b95.353 


-b 159.232 


238.719 


50.813 


NH 4 CIO 4 


s 


117.485 


-295.767 


-88.607 


184.180 


128.072 


CIF 5 


g 


130.4450 


-238.488 


-146.725 


310.739 


97.165 


CIF 3 


g 


92.442 


-158.866 


-118.877 


281.600 


63.845 


N 2 O 4 


1 


92.011 


-19.564 


+97.521 


209.198 


142.509 


N 2 O 4 


g 


92.011 


9.079 


97.787 


304.376 


77.256 


NO 2 


g 


46.0055 


33.095 


51.258 


240.034 


36.974 


HNO 3 


g 


63.0128 


-134.306 


-73.941 


266.400 


53.326 


N 2 


g 


28.0134 


0 


0 


191.609 


29.125 


O 2 


g 


31.9988 


0 


0 


205.147 


29.376 


NH 3 


g 


17.0305 


-45.898 


-16.367 


192.774 


35.652 



“s = solid, 1 = liquid, g = gas. Several species are listed twice, as a liquid and as a gas; the difference 
is due to evaporation or condensation. 

The molar mass can be in g/g-mol or kg/kg-mol and Cp can be in J/g-mol-K or kJ/kg-mol-K. 
Source: Refs. 5-8 and 5-9. 



free energy is defined as G,-; it can be determined for specific thermodynamic 
conditions, for mixtures of gas as well as an individual gas species: 

G^U +pV -TS =h-TS (5-10) 

For most materials used as rocket propellant, the free energy has been deter- 
mined and tabulated as a function of temperature. It can be corrected for pressure. 
G;’s units are J/kg-mol. For a series of different species the mixture free energy 
G is 



j=i 



(5-11) 



164 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



The free energy is a function of temperature and pressure. It is another property 
of a material, just like enthalpy or density; only two such independent parameters 
are required to characterize a gas condition. The free energy may he thought of 
as the tendency or driving force for a chemical material to enter into a chemical 
(or physical) change. Although it cannot he measured directly, differences in 
chemical potential can he measured. When the chemical potential of the reactants 
is higher than that of the likely products, a chemical reaction can occur and the 
chemical composition can change. The change in free energy AG for reactions 
at constant temperature and pressure is the chemical potential of the products 
less that of the reactants: 

m n 

AG = Y1 [«;(A/G‘’);]products - ^ [«; (A/ G°); ] reactants (5-12) 

j=i i=i 

Here the superscript m gives the number of gas species in the combustion 
products, the superscript n gives the number of gas species in the reactants, and 
the AG represents the maximum energy that can be “freed” to do work on an 
“open” system where mass enters and leaves the system. At equilibrium the free 
energy is a minimum; at its minimum a small change in mixture fractions causes 
almost no change in AG, and the free energies of the products and the reactants 
are essentially equal. Then 



dAGIdn = 0 



(5-13) 



and a curve of molar concentration n versus AG would have a minimum. 

If reacting propellants are liquid or solid materials, energy will be needed to 
change phase, vaporize them, or break them down into other gaseous species. This 
energy has to be subtracted from the heat or the energy available to heat the gases 
from the reference temperature to the combustion temperature. Therefore, the val- 
ues of AH^ and AG° for liquid and solid species are different from those of the 
same species in a gaseous state. The standard /ree energy of formation AfG^ is 
the increment in free energy associated with the reaction of forming a given com- 
pound or species from its elements at their reference state. Table 5-2 gives values 
of AfH^ and AfG^ and other properties of carbon monoxide as a function of tem- 
perature. Similar data for other species can be obtained from Refs. 5-7 and 5-13. 
The entropy is another thermodynamic property of matter that is relative, which 
means that it is determined as a change in entropy. In the analysis of isentropic 
nozzle flow, it is assumed that the entropy remains constant. It is defined as 



dU p dV dT dp 

dS = + = C„ R — 

T T T p 



and the corresponding integral is 



5 - 5 ° 






In— - R \n— 
To Po 



(5-14) 



(5-15) 



5.2. ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS 1 65 



TABLE 5-2. Variation of Thermochemical Data with Temperature for Carbon 
Monoxide (CO) as an Ideal Gas 



Temp. 

(K) 


(J/mol-K) 


//® -//®(D 
(kj/mol) 


AfH° 

(kJ/mol) 


A/G® 

(kJ/mol) 


0 


0 


0 


-8.671 


-113.805 


-113.805 


298.15 


29.142 


197.653 


0 


110.527 


-137.163 


500 


29.794 


212.831 


5.931 


-110.003 


-155.414 


1000 


33.183 


234.538 


21.690 


-111.983 


-200.275 


1500 


35.217 


248.426 


38.850 


-115.229 


-243.740 


2000 


36.250 


258.714 


56.744 


-118.896 


-286.034 


2500 


36.838 


266.854 


74.985 


-122.994 


-327.356 


3000 


37.217 


273.605 


93.504 


-127.457 


-367.816 


3500 


37.493 


279.364 


112.185 


-132.313 


-407.497 


4000 


37.715 


284.386 


130.989 


-137.537 


-446.457 



Source: Refs. 5-8 and 5-9. 



where the zero applies to the reference state. In an isentropic process, entropy is 
constant. For a mixture the entropy is 

n 

S =Y^Sfnj (5-16) 

j=i 

Here entropy Sj is in J/kg-mol-K. The entropy for each gaseous species is 

Sj = (S^)j - R In^ -Rlnp (5-17) 

n 

For solid and liquid species the last two terms are zero. Here (S'®) refers to the 
standard state entropy at temperature T. Typical values for entropy are listed in 
Tables 5-1 and 5-2. 



5.2. ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS 

The objectives here are to determine the theoretical combustion temperature 
and the theoretical composition of the resulting reaction products, which in turn 
will allow the determination of the physical properties of the combustion gases 
{Cp,k,p, or other). Before we can make this analysis, some basic data (e.g., 
propellants, their ingredients, desired chamber pressure, or all likely reaction 
products) have to be known or postulated. Although the combustion process 
really consists of a series of different chemical reactions that occur almost simul- 
taneously and includes the breakdown of chemical compounds into intermediate 
and subsequently into final products, the analysis is only concerned with the ini- 
tial and final conditions, before and after combustion. We will mention several 



166 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



approaches to the analysis of chamber conditions. In this section we will first 
give some definitions of key terms and explain some concepts and principles. 

The first principle concerns the conservation of energy. The heat created by the 
combustion is equal to the heat necessary to raise the resulting gases adiabatically 
to their final combustion temperature. The heat of reaction of the combustion 
ArH has to equal the enthalpy change AH of the reaction product gases. The 
energy balance can be thought of as a two-step process. The chemical reaction 
occurs instantaneously but isothermally at the reference temperature, and the 
resulting energy release then heats the gases from this reference temperature to 
the final combustion temperature. The heat of reaction, Equation 5-9, becomes 



Here Ah is the increase in enthalpy for each species multiplied by its molar 
fraction, and Cp is the molar specific heat at constant pressure. 

The second principle is the conservation of mass. The mass of any of the 
atomic species present in the reactants before the chemical reaction must be 
equal to the mass of the same species in the products. This can be illustrated by 
a more general case of the reaction of Equation 5-8. In this case the reactants 
are not in stoichiometric proportion. 

In the combustion of hydrogen with oxygen it is possible to form six products: 
water, hydrogen, oxygen, hydroxyl, atomic oxygen, and atomic hydrogen. In this 
case all the reactants and products are gaseous. Theoretically, there could be two 
additional products: ozone O 3 and hydrogen peroxide H 2 O 2 ; however, these are 
unstable materials that do not readily exist at high temperature, and they can be 
ignored. In chemical notation this can be stated by 

aH 2 + b 02 — > «h2oH20 -|- mh2H2 + no.fi 2 + wqO -H «hH -|- «ohOH (5-19) 

The left side shows the condition before and the right side the condition after 
the reaction. Since H 2 and O 2 can be found on both sides, it means that not all 
of these species are consumed and a portion, namely mh 2 « 02 > will remain 
unreacted. With chemical equilibrium at a particular temperature and pressure 
the molar concentrations on the right side will remain fixed. Here a, b, n^^o, 
nii^,no^,no,ni{, and non are the respective molar quantities of these substances 
before and after the reaction, and they can be expressed in kg-mol per kilogram 
of propellant reactants or reaction products. The initial proportions of a and b are 
usually known. The number of kg-mol per kilogram of mixture of each element 
can be established from this initial mix of oxidizer and fuel ingredients. Eor the 
hydrogen -oxygen relation above, the mass balances would be 




(5-18) 



for hydrogen: 2a = 2wh20 + 2«h2 + «h + «oh 

for oxygen: 2b = «h 20 + 2«02 + «o + «oh 



(5-20) 



5.2. ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS 1 67 



The mass balance of Eq. 5-20 provides two more equations for this reac- 
tion (one for each atomic species) in addition to the energy balance equation. 
There are six unknown product percentages and an unknown combustion or 
equilibrium temperature. However, three equations provide a solution for only 
three unknowns, say the combustion temperature and the molar fractions of two 
of the species. If, for example, it is known that the initial mass mixture ratio of 
b/a is fuel rich, so that the combustion temperature will be relatively low, the 
percentage of remaining O 2 and the percentage of the dissociation products (O, 
H, and OH) would all be very low and can be neglected. Thus no,nn,non, and 
«02 set to be zero. The solution requires knowledge of the enthalpy change 
of each of the species, and that information can be obtained from existing tables, 
such as Table 5-2 or Refs. 5-8 and 5-9. 

In more general form, the mass for any given element must be the same before 
and after the reaction. The number of kg-mol of a given element per kilogram 
of reactants and product is equal, or their difference is zero. For any one atomic 
species, such as the H or the O in Eq. 5-20, 



m 




n 




- 




_;=i 


products 


_;=i 



= 0 



(5-21) 



J propellants 



Here the atomic coefficients ajj are the number of kilogram atoms of element 
i per kg-mol of species j, and m and n are as defined above. The average 
molecular mass of the products from Eq. 5-5 would be 



^ _ 2«h2 + 32«02 + I8WH2O + 16 «o + + 17«oh 

, + no2 + ”H20 + «o + «h + «OH 






(5-22) 



The approach used in Ref. 5-13 is commonly used today for thermochemical 
analysis. It relies on the minimization of the Gibbs free energy and on mass 
balance and energy balance equations. As was explained in Eq. 5-12, the change 
in the Gibbs free energy function is zero at equilibrium (AG = 0): the chemical 
potential of the gaseous propellants has to equal that of the gaseous reaction 
products, which is Eq. 5-12: 



AG = L {Hj a ) products AGj) 

reactants — 0 (5-23) 

To assist in solving this equation a Lagrangian multiplier or a factor of the 
degree of the completion of the reaction is often used. An alternative method 
for solving for the gas composition, temperature, and gas properties is to use the 
energy balance (Eq. 5-18) together with several mass balances (Eq. 5-21) and 
equilibrium constant relationships. 

After assuming a chamber pressure and setting up the energy balance, mass 
balances, and equilibrium relations, one method of solving all the equations is to 
estimate a combustion temperature and then solve for the various values of 



168 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



Then a balance has to be achieved between the heat of reaction A,.H^ and the 
heat absorbed by the gases, Hj — H^, to go from the reference temperature to 
the combustion temperature. If they do not balance, another value of the com- 
bustion temperature is chosen until there is convergence and the energy balances. 

The energy release ejficiency , sometimes called the combustion ejficiency , can 
now be defined as the ratio of the actual change in enthalpy per unit propellant 
mixture to the calculated change in enthalpy necessary to transform the propel- 
lants from the initial conditions to the products at the chamber temperature and 
pressure. The actual enthalpy change can be evaluated if the initial propellant con- 
dition and the actual composition and the temperature of the combustion gases 
are measured. Experimental measurements of combustion temperature and gas 
composition are difficult to perform accurately, and the combustion efficiency is 
therefore actually evaluated only in rare instances. The combustion efficiency in 
liquid propellant rocket thmst chambers depends on the method of injection and 
mixing and increases with increased combustion temperature. In solid propellants 
the combustion efficiency is a function of the grain design, the propellant, and the 
degree of mixing between the several solid constituents. Actual measurements 
on well-designed rocket propulsion systems indicate efficiency values of 94 to 
99%. These high values indicate that the combustion is essentially complete, that 
very little, if any, unreacted propellant remains, and that chemical equilibrium is 
indeed established. 

The number of compounds or species in the exhaust can be 50 or more with 
solid propellants or with liquid propellants that have certain additives. The num- 
ber of nearly simultaneous chemical reactions that have to be considered can 
easily exceed 150. Fortunately, many of these chemical species are present only 
in very small amounts and can usually be neglected. 

Example 5-1. Hydrogen peroxide is used both as a monopropellant and as an oxidizer 
in bipropellant systems. It is stored in liquid form and available in various degrees of 
dilution with liquid water. For rocket applications, concentrations (70 to 98'*'%), known 
as high-test peroxide (HTP) are used. For a monopropellant application, calculate the 
adiabatic flame temperature as a function of water content based on an initial mixture 
temperature of 298.15 K (the standard condition). 

SOLUTION. In this usage, hydrogen peroxide dissociates while passing through a catalyst 
and releases energy which goes to increase the propellant temperature in the absence of 
any heat transfer losses. But some of this heat will be required to evaporate the diluent 
water. The mass balance, Eq. 5-21, is satisfied by 2 mol of hydrogen peroxide in n moles 
of liquid water, producing n + 2 mol of water vapor plus 1 mol of oxygen gas. Since the 
reaction goes to completion, no equilibrium constant is needed: 

2H202(/) -f «H20(/) ^ (n -h 2)H20(g) -f 02(g) 

The symbols (/) and (g) refer to the liquid state and the gaseous state, respectively. 
The heats of formation from the standard state, A///° and molar specific heats Cp are 
shown below (see Table 5-1 and other common sources such as the NIST Chemistry 



5.2. ANALYSIS OF CHAMBER OR MOTOR CASE CONDITIONS 1 69 



web-book, http://webbook.nist.gov/chemistry/). For these calculations the heat of mixing 
may he ignored. 



Species 


AfH° (kJ/kg-mol) 


Cp (J/kg-mol-k) 


9Jl (kg/kg-mol) 


H202(/) 


-187.69 




34.015 


H20(/) 


-285.83 




18.015 


H20(g) 


-241.83 


0.03359 


18.015 


02(g) 


0 


0.02938 


31.999 



The energy balance, Eq. 5-9, for 2 mol of decomposing hydrogen peroxide becomes 

ArH^ = [nAfH°]H20 — [nAfH°]H202 

= 2 X (-241.83) - 2 X (-187.69) 

= -108.28 kJ 



The reaction is exothermic but as stated some of this energy is used up in vaporizing the 
diluent liquid water, namely 285.83 — 241.83 = 44.0 kj/kg-mol (at standard conditions). 
The net available heat release thus becomes 108.28 — 44. On (kJ). In order to calculate 
the adiabatic temperature, we assume ideal-gas heating at constant pressure, Eq. 5-18 
(values for the molar specific heats are from Table 5-1 and are taken as constant): 

j (wH 2 oCpH 20 + MO 2 CPO 2 ) dT = [(2 + n) C;,H 20 + Cp 02 ] AT = 108.28 — 44. On 

It will be more convenient to give results in terms of a mass fraction z of the diluent 
water in the original mixture, and for this the molecular masses need to be inserted (also 
shown in Table 5-1): 

z = niH2o/('«H202 + 'WH 20 ) = n 9J£h2o/(2 971h202 + ” ®Th2o) 

= 18.015n/(2 X 34.015 -b 18.015n) 

Now solve for n 

n =3.78z/(l-z) 

and substitute it in the relation for the temperature. The resulting value (the adiabatic 
temperature. Tad) may then he plotted in terms of the mass fraction z with the initial 
temperature as given: 

Tad = [108.28 - 44 X 3.78z/(l - z)]/[0.03359 x (2 -b 3.78z/(l - z)) 

-b 0.02838] -b 298. 15d 

The figure also displays the values of c* (where T\ = Tad), 971, and k, which are 
calculated according to Eqs. 3-32 and 5-5 and 5-7. 



170 



CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 




5.3. ANALYSIS OF NOZZLE EXPANSION PROCESSES 

There are several methods for analyzing the nozzle flow, depending on the 
assumptions made for chemical equilibrium, nozzle expansion, particulates, or 
energy losses. Several are outlined in Table 5-3. 

Once the gases reach the nozzle, they experience an adiabatic, reversible 
expansion process which is accompanied by a major drop in temperature and 
pressure and a conversion of thermal energy into kinetic energy. Several increas- 
ingly more complicated methods have been used for the analysis of the process. 
For the simple case of frozen equilibrium and one-dimensional flow the state of 
the gas throughout expansion in the nozzle is fixed by the entropy of the sys- 
tem, which is presumed to be invariant as the pressure is reduced to the value 
assigned to the nozzle exit plane. All the assumptions listed in Chapter 3 for an 
ideal rocket would be valid here. Again, the effects of friction, divergence angle, 
heat losses, shock waves, or nonequilibrium are neglected in the simple cases 
but are considered in the more sophisticated solutions. The condensed (liquid or 
solid) phases are similarly assumed to have zero volume and to be in kinetic 
as well as thermal equilibrium with the gas flow. This implies that particles or 
droplets are very small in size, move at the same velocity as the gas stream, and 
have the same temperature as the gas at all places in the nozzle. 

The chemical equilibrium during expansion in the nozzle can be analytically 
regarded in the following ways: 

1. When the composition is invariant throughout the nozzle, there are no 
chemieal reactions or phase changes and the product composition at the 
nozzle exit is identical to that of its chamber condition. The results are 
known as frozen equilibrium rocket performance. This method usually is 
simple, but underestimates the performanee, typically by 1 to 4%. 



5.3. ANALYSIS OF NOZZLE EXPANSION PROCESSES 



171 



2. Instantaneous chemical equilibrium among all molecular species is main- 
tained under the continuously variable pressure and temperature conditions 
of the nozzle expansion process. Thus the product composition shifts; sim- 
ilarly, instantaneous chemical reactions, phase changes or equilibria occur 
between gaseous and condensed phases of all species in the exhaust gas. 
The results so calculated are called shifting equilibrium performance. The 
gas composition mass percentages are different in the chamber and the 
nozzle exit. This method usually overstates the performance values, such 
as c* or /j, typically by 1 to 4%. Here the analysis is more complex. 

3. Chemical reactions do not occur instantaneously; even though the reac- 
tions occur rapidly, they require a finite time. Reaction rates of specific 
reactions can be estimated; the rates are usually a function of tempera- 
ture, the magnitude of deviation from the equilibrium molar composition, 
and the nature of the chemicals or reactions involved. The values of T, 
c*, or Is for these types of equilibrium analysis usually are between those 
of frozen and instantaneously shifting equilibria. This approach is almost 
never used because of the lack of good data on reaction rates with multiple 
simultaneous chemical reactions. 



For an axisymmetric nozzle, both one-and two-dimensional analyses can be 
used. The simplest nozzle flow analysis is one dimensional, which means that 
all velocities and temperatures or pressures are equal at any normal cross section 
of an axisymmetric nozzle. It is often satisfactory for preliminary estimates. In a 
two-dimensional analysis the velocity, temperature, density, and/or Mach number 
do not have a flat profile and vary somewhat over the cross sections. For nozzle 
shapes that are not bodies of revolution (e.g., rectangular, scarfed, or elliptic) a 
three-dimensional analysis should be performed. 

If solid particles or liquid droplets are present in the nozzle flow and if the 
particles are larger than about 0.1 pm average diameter, there will be a thermal 
lag and velocity lag. The solid particles or liquid droplets do not expand like 
a gas; their temperature decrease depends on losing energy by convection or 
radiation, and their velocity depends on the drag forces exerted on the particle. 
Larger-diameter droplets or particles are not accelerated as rapidly as the smaller 
ones and flow at a velocity lower than that of the adjacent accelerating gas. Also, 
the particulates are hotter than the gas and provide heat to the gas. While these 
particles contribute to the momentum of the exhaust mass, they are not as efficient 
as an all-gaseous exhaust flow. For composite solid propellants with aluminum 
oxide particles in the exhaust gas, the loss due to particles could typically be 1 to 
3%. The analysis of a two- or three-phase flow requires knowledge of an assump- 
tion about the nongaseous matter, the sizes (diameters), size distribution, shape 
(usually assumed to be spherical), optical surface properties (for determining the 
emission/absorption or scattering of radiant energy), and their condensation or 
freezing temperatures. Some of these parameters are not well known. Performance 
estimates of flows with particles are explained in Section 3.5. 



1 72 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



TABLE 5-3. Typical Steps and Alternatives in the Analysis of Rocket Thermochemical 
Processes in Nozzles 



Step 



Process Method/Implication/Assumption 



Nozzle inlet 
condition 

Nozzle 

expansion 



Same as chamber exit; need to 
know T\,pi,v\,H ,c* , p\, etc. 

An adiabatic process, where flow 
is accelerated and thermal 
energy is converted into kinetic 
energy. Temperature and 
pressure drop drastically. 

Several different analyses have 
been used with different specific 
effects. Can use one-, two-, or 
three-dimensional flow pattern. 



For simpler analyses assume the 
flow to be uniformly mixed and 
steady. 

1 . Simplest method is inviscid isen- 
tropic expansion flow with con- 
stant entropy. 

2. Include internal weak shock 
waves; no longer a truly isentropic 
process. 

3. If solid particles are present, they 
will create drag, thermal lag, 
and a hotter exhaust gas. Must 
assume an average particle size 
and optical surface properties of 
the particulates. Flow is no longer 
isentropic. 

4. Include viscous boundary layer 
effects and/or nonuniform veloc- 
ity profile. 



Often a simple single correction factor is used with one-dimensional analyses to 
correct the nozzle exit condition for items 2, 3, and/or 4 above. Computational fluid 
dynamic codes with finite element analyses have been used with two- and 
three-dimensional nozzle flow. 



Chemical 

equilibrium 

during 

nozzle 

expansion 



Due to rapid decrease in T and p, \ 
the equilibrium composition can 
change from that in the 
chamber. The four processes 
listed in the next column allow 
progressively more realistic 
simulation and require more 
sophisticated techniques. 



Frozen equilibrium; no change in 
gas composition; usually gives 
low performance. 

2. Shifting equilibrium or instan- 
taneous change in composition; 
usually overstates the perfor- 
mance slightly. 

3. Use reaction time rate analysis to 
estimate the time to reach equi- 
librium for each of the several 
chemical reactions; some rate 
constants are not well known; 
analysis is more complex. 

4. Use different equilibrium analysis 
for boundary layer and main invis- 
cid flow; will have nonuniform 
gas temperature, composition, and 
velocity profiles. 



5.3. ANALYSIS OF NOZZLE EXPANSION PROCESSES 1 73 



TABLE 5-3. (Continued) 

Step Process Method/Implication/Assumption 



Heat release in 
nozzle 



Nozzle shape 
and size 



Gas properties 



Nozzle exit 
conditions 



Calculate 

specific 

impulse 



Recombination of dissociated 
molecules (e.g., H + H = H 2 ) 
and exothermic reactions due to 
changes in equilibrium 
composition cause an internal 
heating of the expanding gases. 
Particulates release heat to the 
gas. 

Can use straight cone, bell-shaped, 
or other nozzle contour; bell 
can give slightly lower losses. 
Make correction for divergence 
losses and nonuniformity of 
velocity profile. 



The relationships governing the 
behavior of the gases apply to 
both nozzle and chamber 
conditions. As gases cool in 
expansion, some species may 
condense. 

Will depend on the assumptions 
made above for chemical 
equilibrium, nozzle expansion, 
and nozzle shape/contour. 
Assume no jet separation. 
Determine velocity profile and 
fhe pressure profile af the 
nozzle exit plane. If pressure is 
not uniform across a section it 
will have some cross flow. 



Can be determined for different 
altitudes, pressure ratios, 
mixture ratios, nozzle area 
ratios, etc. 



Heat released in subsonic portion 
of nozzle will increase the exit 
velocity. Heating in the 
supersonic flow portion of 
nozzle can increase the exit 
temperature but reduce the exit 
Mach number. 

Must know or assume a particular 
nozzle configuration. Calculate 
bell contour by method of 
characteristics. Use Eq. 3-34 
for divergence losses in conical 
nozzle. Most analysis programs 
are one- or two-dimensional. 
Unsymmetrical nonround 
nozzles may need 
three-dimensional analysis. 

Either use perfect gas laws or, if 
some of the gas species come 
close to being condensed, use 
real gas properties. 

Need to know the nozzle area 
ratio or nozzle pressure ratio. 
For quasi-one-dimensional and 
uniform nozzle flow, see Eqs. 
3-25 and 3-26. If V 2 is not 
constant over the exit area, 
determine effective average 
values of V 2 and p 2 - Then 
calculate profiles of T, p, efc. 
For nonuniform velocify profile, 
fhe solufion requires an iterative 
approach. Can calculate the gas 
conditions (T, p, etc.) at any 
point in the nozzle. 

Can be determined for average 
values of V 2 ,P 2 , and p^ based 
on Eqs. 2-6 or 2-14. 



1 74 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



The viscous boundary layer next to the nozzle wall has velocities substantially 
lower than that of the inviscid free stream. The slowing down of the gas flow near 
the wall due to the viscous drag actually causes the conversion of kinetic energy 
back into thermal energy, and thus some parts of the boundary layer can be hotter 
than the local free-stream static temperature. A diagram of a two-dimensional 
boundary layer is shown in Figure 3-16. With turbulence this boundary layer 
can be relatively thick in large-diameter nozzles. The boundary layer is also 
dependent on the axial pressure gradient in the nozzle, the nozzle geometry, 
particularly in the throat region, the surface roughness, or the heat losses to the 
nozzle walls. Theoretical boundary layer analyses with unsteady flow are only 
approximations, but are expected to improve in the future as our understanding of 
the phenomena and computational fluid dynamics (CFD) techniques are validated. 
The net effect is a nonuniform velocity and temperature profile, an irreversible 
friction process in the viscous layers, and therefore an increase in entropy and a 
slight reduction (usually less than 5%) of the kinetic exhaust energy. The layers 
immediately adjacent to the nozzle walls have laminar and subsonic flow. 

At the high combustion temperatures a small portion of the combustion gas 
molecules dissociate (split into simpler species); in this dissociation process some 
energy is absorbed. When energy is released during reassociation (at lower pres- 
sures and temperatures in the nozzle), this reduces the kinetic energy of the 
exhaust gas at the nozzle exit. This is discussed further next. 

For propellants that yield only gaseous products, extra energy is released in 
the nozzle, primarily from the recombination of free-radical and atomic species, 
which become unstable as the temperature is decreased in the nozzle expansion 
process. Some propellant products include species that condense as the tem- 
perature drops in the nozzle expansion. If the heat release on condensation is 
large, the difference between frozen and shifting equilibrium performance can be 
substantial. 

In the simplest method the exit temperature T 2 is determined for an isen- 
tropic process (frozen equilibrium) by considering the entropy to be constant. 
The entropy at the exit is the same as the entropy in the chamber. This deter- 
mines the temperature at the exit and thus the gas condition at the exit. From 
the corresponding change in enthalpy it is then possible to obtain the exhaust 
velocity and the specific impulse. For those analysis methods where the nozzle 
flow is not really isentropic and the expansion process is only partly reversible, 
it is necessary to include the losses due to friction, shock waves, turbulence, and 
so on. The result is a somewhat higher average nozzle exit temperature and a 
slight loss in /^ . A possible set of steps used for the analysis of nozzle processes 
is given in Table 5-3. 

When the contraction between the combustion chamber (or the port area) and 
the throat area is small (Ap/A, < 3), the acceleration of the gases in the chamber 
causes a drop in the effective chamber pressure at the nozzle entrance. This 
pressure loss in the chamber causes a slight reduction of the values of c and 4 . 
The analysis of this chamber configuration is treated in Ref. 5-14 and some data 
are briefly shown in Table 3-2. 



5.4. COMPUTER-ASSISTED ANALYSIS 175 



5.4. COMPUTER-ASSISTED ANALYSIS 

All the analyses discussed in this chapter are done today with computer programs. 
Most are based on minimizing the free energy. This is a simpler approach than 
relying on equilibrium constants, which was used some years ago. Once the 
values of nj and Ti are determined, it is possible to calculate the molecular mass 
of the gases (Eq. 5-5), the average molar specific heats Cp by a similar formula, 
and the specific heat ratio k from Eqs. 3-6 and 5-7. This then characterizes 
the thermodynamic conditions in the combustion chamber. With these data we 
can calculate c*, R, and other mixture parameters of the chamber combustion. 
The nozzle expansion process simulated with computers give the performance 
(such as Is, c, or A 2 /A,) and the gas conditions in the nozzle; it usually includes 
several of the corrections mentioned in Chapter 3. Programs exist for one-, two-, 
and three-dimensional flow patterns. 

More sophisticated solutions include a supplementary analysis of combustion 
chamber conditions where the chamber velocities are high (see Ref. 5-14), a 
boundary layer analysis, a heat transfer analysis, or a two-dimensional axisym- 
metric flow with nonuniform flow properties across any cross section of the 
nozzle. Time-dependent chemical reactions in the chamber are usually neglected, 
but they can be analyzed by estimating the time rate at which the reaction occurs. 
This is described in Ref. 5-3. 

A commonly used computer program, based on equilibrium compositions, 
has been developed at the NASA Glenn Laboratories and is known as the 
NASA CEA code (Chemical Equilibrium with Applications). It is described in 
Ref. 5-13, Vols. 1 and 2. Key assumptions for this program are one-dimensional 
forms of the continuity, energy and momentum equations, negligible velocity at 
the forward end of the combustion chamber, isentropic expansion in the noz- 
zle, ideal gas behavior, and chemical equilibrium in the combustion chamber. It 
includes options for frozen flow and for narrow chambers (for liquid propellant 
combustion) or port areas with small cross sections (for solid propellant grains), 
where the chamber flow velocities are relatively high, resulting in extra pressure 
losses and a slight loss in performance. NASA’s CEA code has recently become 
part of a commercial code named Cequel™, which extends the code’s original 
capabilities. 

Other relatively common computer codes used in the United States for ana- 
lyzing the converging-diverging nozzle flow include: 

ODE (one-dimensional equilibrium code) which features infinitely fast chem- 
ical reactions (shifting equilibrium) and includes all gaseous constituents. 

ODK (one-dimensional kinetics) which incorporates finite chemical reaction 
kinetic rates for temperature-dependent composition changes in the flow 
direction with uniform flow properties across any nozzle section. It is used 
as a module in more complex codes and has no provision for particles. 

TDK (two-dimensional kinetic code) which incorporates finite kinetic chemical 
reaction rates and radial variance in flow properties. It has no provision for 
particles. 



1 76 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



VIPERP ( viscous interaction performance evaluation routine for two-phased 
flows) a parabolized Navier-Stokes code for internal two-phased nozzle 
flow with turbulent and nonequilibrium reacting gases. It can be used with 
solid particles but requires data (or assumptions) on the amount of solids, 
particle size distribution, or their shape (see, e.g., pp. 503 to 505 in 7th 
Edition). 

More information on these computer codes may be obtained from the appro- 
priate government offices and/or from small companies (who actually run the 
nozzle codes for their customers). Some of the more sophisticated codes are 
proprietary to propulsion organizations and not publicly available. 



5.5. RESULTS OF THERMOCHEMICAL CALCULATIONS 

Extensive computer results of these machine calculations are available and only a 
few samples are indicated here to illustrate typical effects of the variations of key 
parameters. In general, high specific impulse or high values of c* can be obtained 
if the average molecular mass of the reaction products is low (usually this implies 
a formulation rich in hydrogen) and/or if the available chemical energy (heat of 
reaction) is large, which means high combustion temperatures (see Eq. 3-16). 

Table 5-4 shows calculated results for a liquid oxygen, liquid hydrogen thrust 
chamber taken from an example of this reference. It has shifting equilibrium in 
the nozzle flow. The narrow chamber has a cross section that is only a little 
larger than the throat area. The large pressure drop in the chamber (approximately 
126 psi) is due to the energy needed to accelerate the gas, as discussed in Section 
3.3 and Table 3-2. 

Values of calculated specific impulse will be higher than those obtained from 
firing actual propellants in rocket units. In practice it has been found that the 
experimental values are about 3 to 12% lower than those calculated by the 
method explained in this chapter. Because the nozzle inefficiencies explained in 
Chapter 3 must be considered, only a portion of this correction (perhaps 1 to 
4%) is due to combustion inefficiencies. 

Eigures 5-1 to 5-6 indicate the results of performance calculations for the liq- 
uid propellant combination, liquid oxygen-RP-1. These data are taken from Refs. 
5-7 and 5-8. The RP-1 fuel is a narrow-cut hydrocarbon similar to kerosene 
with an average of 1.953 mol of hydrogen for each mole of carbon; thus it has 
a nominal formula of CHi 953 . The calculations are limited to a chamber pres- 
sure of 1000 psia. Most of the curves are for optimum area ratio expansion to 
atmospheric pressure, namely 1 atm or 14.696 psia, and for a limited range of 
oxidizer-to-fuel mixture ratios. 

Eor maximum specific impulse. Figs. 5-1 and 5-4 show an optimum mixture 
ratio of approximately 2.3 (kg/sec of oxidizer flow divided by kg/sec of fuel 
flow) for frozen equilibrium expansion and 2.5 for shifting equilibrium with 
gas expansion to sea-level pressure. The maximum values of c* are at slightly 



0^ On 

O -H — 

r- ra 



o r- ON 
lO 00 o 

'Tf <T) 



OOCNOOOOOO 

OOiO'TfOOOO 

OOOOnOOOO 

OOOOnOOOO 

OOrONOOOOO 

doddoodo 



c 

.2 

!/3 

c 

03 

Oh 

X 

PJ 



X 



(N O 
<N <N ra 00 o 



(N 00 (T> 
00 NO — 
«N 



(NOOOOOOOO 

OOiO'^fOOOO 

OOOONOOOO 

OOOONOOOO 

oomNOOOOO 

dddddddd 



O 

Z 

d 

D 



d 

o 

PP 



(N (N 
NO d 



r- ON 

(N IT) 00 (N 
‘ CN lO 



'^or-iooo'^o 

(Nommoooo 

OOOONOOOO 

OOOONOOOO 

OOrONOOOOO 

dddddddd 



o 

o 



bO 

c 

PJ 



D NO r-- 

15 r-- 00 o «N 

X O — ON — H (N 

W doNco — dddincN 



On cm 
lO lO 
NO ON 
On 00 
(N NO 



o ON r- ON 
o o r- o 
0 0^0 
o o o o 
o o o o 



oooooooo 



o 

a 

D 

biD 

o 

u, 

T3 

K 

IB 

'B 

cr 

T3 

C 

cd 

d 

D 

bO 

X 

O 



d 

O' 



•c 

X) 



X 



o 

H 






o o o 
lo CO o o 
00 — lo O O 



Co 



On r-- 
r- <T) 
00 IT) 



X 

B 

o 

u 



— 00 O — 



C-; — <N p 
On <N — 00 d X 



00 o o 
lO m ‘ 

ro m 
On IT) 

(N X 



O O (N O 
O O O O 



S300000000 



00 
00 lo 
m 00 
On m 
(N X 



O (N O -H 
O O ro O 

o o o o 
dddddddd 



o rs o CO 

On O 'Tf 

ro O X 
ro O On ro 
O O (N X 



O O ro O 
o o o o 



oooooooo 



d 

U 



PP 

nP 

PP 



'2 ^ 
C3 ^ 
<U 



I 



X 

U 



• ^ ^ 
^ _g 

d § 

^ B 

X c 



'X, 8 

s; t-H 






(U c/5 

C« 



^ -j; o 3 



O « 
S ffi 



o 

' s 



K N 
O O 



177 



‘'Chamber contraction ratio Ai/A,. 

*If cut off at throat. 

c is the effective exhaust velocity in a vacuum. 

V 2 is the nozzle exit velocity at optimum nozzle expansion. 
NA means not applicable. 



178 



CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



1850- 
S 1800- 

I/) 

E 

o 1750- 
1700- 




Mixture ratio (oxidizer/fuel) 



FIGURE 5-1. Calculated performance analysis of liquid oxygen and hydrocarbon fuel 
as a function of mixture ratio. 



different mixture ratios. This optimum mixture ratio is not the value for highest 
temperature, which is usually fairly close to the stoichiometric value. The stoi- 
chiometric mixture ratio is more than 3.0 where much of the carbon is burned to 
CO 2 and almost all of the hydrogen to H 2 O. 

Because shifting equilibrium makes more enthalpy available for conversion to 
kinetic energy, it gives higher values of performance (higher Is or c*) and higher 
values of nozzle exit temperature for the same exit pressure (see Fig. 5-1). The 
influence of mixture ratio on chamber gas composition is evident from Fig. 5-2. 
A comparison with Fig. 5-3 indicates the marked changes in the gas composition 
as the gases are expanded under shifting equilibrium conditions. The influence of 
the degree of expansion, or of the nozzle exit pressure on the gas composition, 
is shown in Fig. 5-6. As the gases are expanded to higher area ratios and lower 
exit pressure (or higher pressure ratios) the performance increases; however. 



5.5. RESULTS OF THERMOCHEMICAL CALCULATIONS 179 




FIGURE 5-2. Calculated chamber gas composition for liquid oxygen and hydrocarbon 
fuel as a function of mixture ratio. Aggressive gases, such as O 2 , O, or OH, can cause 
oxidation of the wall materials In the chamber and the nozzle. 



the relative increase diminishes as the pressure ratio is further increased (see 
Figs. 5-5 and 5-6). 

Dissociation of molecules absorbs considerable energy and causes a decrease 
in the combustion temperature, which in turn can reduce the specific impulse. Dis- 
sociation of the reaction products increases as the chamber temperature rises, and 
decreases with increasing chamber pressure. Atoms or radicals such as monatomic 
O or H and OH are formed, as can be seen from Fig. 5-2; some unreacted O 2 also 
remains at the higher mixture ratios and very high combustion temperatures. As 
the gases are cooled in the nozzle expansion, the dissociated species react again 
to form molecules and release heat into the flowing gases. As can be seen from 
Fig. 5-3, only a small percentage of dissociated species persists at the nozzle 
exit and only at the high mixture ratio, where the exit temperature is relatively 
high. (See Fig. 5-1 for exit temperatures with shifting equilibria). Heat released 
in a supersonic flow actually reduces the Mach number. 



180 



CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 




Mixture ratio (oxidizer/fuelt 

FIGURE 5-3. Calculated nozzle exit gas composition for shifting equilibrium conditions 
as a function of mixture ratio. Breakdown into O, OH, or H and free O 2 occurs only at 
the higher temperatures or higher mixture ratios. 



Results of calculations for several different liquid and solid propellant combi- 
nations are given in Tables 5-5 and 5-6. For the liquid propellant combinations, 
the listed mixture ratios are optimum and their performance is a maximum. For 
solid propellants, practical considerations (such as propellant physical properties) 
do not always permit the development of a satisfactory propellant grain when the 
ingredients are mixed in optimum performance proportions (insufficient binder); 
therefore the values listed for solid propellants in Table 5-6 correspond in part 
to practical formulations with reasonable physical and ballistic properties. 

Calculated data obtained from Ref. 5-13 are presented in Tables 5-7 to 5-9 
for a specific solid propellant to indicate typical variations in performance or 
gas composition. This particular propellant consists of 60% ammonium perchlo- 
rate (NH 4 CIO 4 ), 20% pure aluminum powder, and 20% of an organic polymer 
of an assumed chemical composition, namely C 3 . 1 ON 0 . 84 FI 5 Table 5-7 shows 



5.5. RESULTS OF THERMOCHEMICAL CALCULATIONS 181 




1.8 2.0 2.2 2.4 2.6 2.8 3.0 

Mixture ratio (oxidizer/fuel) 

FIGURE 5-4. Variation of theoretical specific impulse with mixture ratio and pressure 
ratio, calculated for frozen equilibrium. 



the variation of several performance parameters with different chamber pressures 
expanding to atmospheric exit pressure. The area ratios listed are optimum for 
this expansion with shifting equilibrium. The exit enthalpy, exit entropy, thrust 
coefficient, and the specific impulse also consider shifting equilibrium conditions. 
The characteristic velocity c* and the chamber molecular mass are functions of 
chamber conditions only. Table 5-8 shows the variation of gas composition with 
chamber pressure. Some of the reaction products are in the liquid phase, such 
as AI 2 O 3 . Table 5-9 shows the variation of nozzle exit characteristics and com- 
position for shifting equilibria as a function of exit pressure or pressure ratio 
for a fixed value of chamber pressure. Table 5-9 shows how the composition is 
shifted during expansion in the nozzle and how several of the species present in 
the chamber have disappeared at the nozzle exit. These three tables show theo- 
retical results calculated on a computer; some of the thermodynamic properties 
of the reactants and reaction products probably do not warrant the indicated high 
accuracy of five significant figures. In the analysis for chemical ingredients of this 
solid propellant, approximately 76 additional reaction products were considered 



002 ^ I— 1 — I — I 1 i 1 M 1 1 1 — I Mill I I I 1 I 1 I I I I ozfr 



)U3pi|^ao3 )snjqx 

^ 00 CM vp o 

CM ^ ^ o o 

I — ^ ^ 1 

>1. ‘ajnjejadujax 




D8S ’8S|ndLU! ojjpeds 

I I I I I rT T I I 1 — I — I 1 1 

8 8SS§ S oooio^ cv, 



0|)ej eajy 



182 



FIGURE 5-5. Variation of calculated parameters with pressure ratio for liquid oxygen-hydrocarbon propellant at a mixture ratio of 2 . 20 . An 
increase in pressure ratio is due to an increase in chamber pressure, a decrease of nozzle exit pressure (larger area ratio and higher altitude), or 
both. 




)ua3J3d 3|oai ‘uoDisoduioQ 



183 



FIGURE 5-6. Variation of exhaust gas composition at nozzle exit with pressure ratio at a fixed mixture ratio and for shifting equilibrium. For 
frozen equilibrium the composition would be the same as in the chamber, as shown in Fig. 5-2. 



TABLE 5-5. Theoretical Performance of Liquid Rocket Propellant Combinations 



Xi 

B • 

c3 

X 

U 



X * 

B D. 

X S 

6 



2 '5 

O 0) 

<C 00 



O 



CQ — 



>» 

CQ 



O lO 
O Oh 
ro (N 



lo Oh 
X 00 

m m 



00 00 CN 

r- lo r- 

fNl (N (N 



lO 

Oh IT) 



mroOhOOhrooomiO'^OhooioOhOhO' 



lom — CN00(N’^OiO'^00 00'^0hin(N(N — 
roiot^OhCNmr^om\0(NOm'^hOooir) — 
oooooooo'Tf'^r^oooooo-HiNioior^r^hor^ 

— — -h(N(N(N(N — -H — — 



I fo hO oo hO 



^ Oh fs -H 

Oh Oh O 00 o 

r- in in ho ho r- 



hOhOin’^OhOh-Hr-'(N'^rofnoooo(NCN<NOhooinor--(NO 
cscMoooinOhr^r^'^Ohin — oooinin"^r^OhOhOr^mOhr^ho 
inin(N’^OhOhinhOinininc^oOhfNi-H(Ncn(Nfn(N-H(NOh-Hr^ 

mmfnm(N(Nmrofnm'^'^mrocnmmcnromcnmmmm<N 



^Ohor^hooo^(NhOooOh — minoraoo — mohoincomr^oh 
ooooop<N(NppOhOh(Nfnrn'^fNl<N— -CNlcNCN^rncncacNC^ 
oo — ^c30^-^oo— 'c30^-^— 



' O O O O ' 



' O O O O ' 



oO'^oo<N’^hC)OhinroO'^ooo'^(NO inin mo 
(Npc^p'^pcNinmppminppmpp'^t— -p-Hoqr-(NO 
mmoom-^c^tN — -H— 



N 


<u 




X 


'n 


a 

Urn 


O 








•o 


'O 


cL 


Q 


>> 


X 


X 


X 


D 





>> 

K 



>, 

K 



a-s 

X N 

S S 

Q?, 

Os: 

^ ^ 2 S 



On 



-I 



^ ^ o' 

in in 05 



X 

O 



S o 



V. 'c 






00^ 



>. ' 

X 



184 



Notes: 

Combustion chamber pressure — 1000 psia (6895 kN/m^); nozzle exit pressure — 14.7 psia (1 atm); optimum expansion. 
Adiabatic combustion and isentropic expansion of ideal gas. 

The specific gravity at the boiling point was used for those oxidizers or fuels that boil below 20°C at 1 atm pressure. 
Mixture ratios are for approximate maximum value of Is . 



5.5. RESULTS OF THERMOCHEMICAL CALCULATIONS 185 



TABLE 5-6. Theoretical Performance of Typical Solid Rocket Propellant Combinations 



Oxidizer 


Fuel 


P 

(g/cm^)“ 


Ti 

(K) 


C* 

(m/sec)* 


(kg/mol) 


(sec)* 


k 


Ammonium 

nitrate 


1 1 % binder and 
7% additives 


1.51 


1282 


1209 


20.1 


192 


1.26 


Ammonium 

perchlorate 

78-66% 


18% organic 
polymer binder 
and 4-20% 
aluminum 


1.69 


2816 


1590 


25.0 


262 


1.21 


Ammonium 

perchlorate 

84-68% 


12% polymer 
binder and 
4-20% 
aluminum 


1.74 


3371 


1577 


29.3 


266 


1.17 



“Average specific gravity of solid propellant. 

^Conditions for f and c* : Combustion chamber pressure: 1000 psia; nozzle exit pressure: 14.7 
psia; optimum nozzle expansion ratio; frozen equilibrium. 



in addition to the major product species. This includes, for example, CN, CH, 
CCl, Cl, NO, and so on. Their calculated mole fractions were very small and 
therefore they have been neglected and are not included in Table 5-8 or 5-9. 

Calculations of this type are useful in estimating performance {Is,c*,Cf,e, 
etc.) for a particular chamber pressure and nozzle exit pressure, and knowl- 
edge of the gas composition, as indicated by the previous figures and tables, 
permits a more detailed estimate of other design parameters, such as gas-hlm 
properties for heat transfer determination, radiation characteristics of the flame 
inside and outside the thrust chambers, and the acoustic characteristics of the 
gases. Performance data relevant to hybrid propellants are presented briefly in 
Chapter 16. 

The thermochemical analysis of this chapter can also be applied to gas gener- 
ators; the results (such as gas temperature Ti, the specific heat Cp, specific heat 
ratio k, or composition) are used for estimating turbine inlet conditions or tur- 
bine power. In gas generators and prebumers for staged combustion cycle rocket 
engines (explained in Section 6.6) the gas temperatures are much lower, to avoid 
damage to the turbine blades. Typically, the combustion reaction gases are at 800 
to 1200 K, which is lower than the gas in the thrust chamber (2900 to 3600 K). 
Examples are listed in Table 5-10 for a chamber pressure of 1000 psia. Some 
species in the gases will not be present (such as atomic oxygen or hydroxyl), 
and often real gas properties will need to be used because some of these gases 
do not behave as a perfect gas at these temperatures. 



c 

D 

Ph 



D 

Q- 

O 

u 

cu 

D 

u 

2 



< O' 

c ® 



s: 
fc ^ 

a> 43 



c3 ^ 

Oh a 

U td 

0 

§ - 

1 "p 

£ s 

u -> 

^ g 
S 2 

’H c 

o 

‘cfl 

u g 

f I 2 

o ^ 
® X 
C PJ 

.9 Vh 

5 (u 
S 



r- p 

I £ 

M (U 

H U 



CN 





o\ 










o 


00 


00 




r- 








o 


Ov 




o 






i> 




CN 


o 




<N 


CN 








o 


CN 




'O 


I> 






CN 


CO 


O) 


VO 


CO 










CO 


»o 


o 


ro 


I> 


CO 


CN 




r4 


CO 




c4 




CN 


CO 


r- 


vd 




o 




o 


CO 


r- 


r- 




cs 


CN 








o 


1— 1 






CN 




(N 




»o 


o 












CN 


CN 


»o 










m 


CN 


1 

























CO 





CO 










l> 


O 


CO 














i> 




CN 






r- 




»o 


»o 


VO 


Ov 


CO 








CO 


Ov 




q 


VO 


q 




00 


CM 


o 


I> 


oo 




l> 


CO 




in 


in 


o 




vd 


i> 


oi 


GTn 


CM 


Ov 


o\ 


CM* 




>o 


l> 


»o 


cd 




o 


CO 


I> 


CN 


r- 


1-H 




CM 


CM 






VO 


CO 


CM 






»o 




CN 


CO 


»o 


CM 












CM 


CM 


»o 










CO 


CN 


1 

























i> r- 















Ov 


Ov 


O 




CO 










VO 




CO 






r- 


CM 


CM 




CM 


VO 


CO 








o 


ON 




q 


CN 


CO 






CM 




00 


00 




q 


q 




q 


q 


o 






vd 


CM 


CM 


CM* 


Ov 


ON 


CM* 






cd 


r- 


00 




in 


»n 


o 


CM 


r- 


00 




CM 


CM 






o 




CM 






i> 




CO 


CM 


in 


CM 












CM 


CM 


»n 










CO 


CM 


1 



























VO 










o 


»n 


CO 




»n 




















r- 


in 


1— 




in 


O 


CO 














q 




q 




-H 


CM 


CM 


00 


q 






o 




in 


q 


O 


00 


CM 


in 


CM 


in 


CM* 


ON 


ON 


cd 




d 


vd 


ON 


d 




O 


VO 


CM 


CO 


r- 


CM 




CM 


CM 






00 


in 


CM 


1-H 




o 




CO 


<— 


»n 


CO 












CM 


CM 


»n 










CO 


CM 


1 

























'sO o^ 















CM 


CO 


ON 




VO 










O 




r- 






r- 


On 


00 


o 


I> 


o 


CO 








ON 


o 




q 


ON 


l> 








CO 


00 


CM 






q 




CM 


l> 


o 


CM 


vd 


l> 


CM 


CM 


CM* 


ON 


ON 


cd 






in 


CM 






o 


o 




o 


r- 


00 




CM 


CM 






00 


VO 


CO 






in 




CO 


o 


in 


CO 












CM 


CM 


»n 










CO 


CM 


1 

























a, 

o 









2 c 

III 

D a (D 

Cfl Q. 
V3 Cfl ^ 

D (U 5 
^ ^ 

G< Ph ii 

u Vh ;-H 
D (U (U 
43 43 43 

see 

cd cd 

43 4= -C 

u u u 



D cd 

s ^ 

u 

D >, 
Oh a 
E cd 



00 

4^ 



o 

6 

oi) 

4^ 



0^3 
^ Po o 
^ 3 ® 

43 ^ 

^ (U 

i S 's 

.32 i3 cd 
X C 43 

PJ PP u 



h ^ 

p !Z3 

B ^ 

I = 

y 3 
1 > 



p o 
o ^ 
S S 

'S '?< 

w w 



c 

^o 

V3 

C ^ 
C4 -i? 

X P 
s 



5 'S 

U 



D 

Dh 



D c. 
D- -.3 



S E 



o o 
’o 'o 

dJ d) 

a Oh 
CO CO 



D 2 
Dh ^ 

CO U 




< 

>3 



186 



Source: From Ref. 5-13. 



5.5. RESULTS OF THERMOCHEMICAL CALCULATIONS 187 



TABLE 5-8. Mole Fraction Variation of Chamber Gas Composition with Combustion 
Chamber Pressure for a Solid Propellant 



Pressure (psia) 


1500 


1000 


750 


500 


200 


Pressure (atm) or 


102.07 


68.046 


51.034 


34.023 


13.609 


pressure ratio 
Ingredient 
A1 


0.00007 


0.00009 


0.00010 


0.00012 


0.00018 


AlCl 


0.00454 


0.00499 


0.00530 


0.00572 


0.00655 


AlClz 


0.00181 


0.00167 


0.00157 


0.00142 


0.00112 


AICI3 


0.00029 


0.00023 


0.00019 


0.00015 


0.00009 


AlH 


0.00002 


0.00002 


0.00002 


0.00002 


0.00002 


AlO 


0.00007 


0.00009 


0.00011 


0.00013 


0.00019 


AlOCl 


0.00086 


0.00095 


0.00102 


0.00112 


0.00132 


AlOH 


0.00029 


0.00032 


0.00034 


0.00036 


0.00041 


AIO2H 


0.00024 


0.00026 


0.00028 


0.00031 


0.00036 


AI2O 


0.00003 


0.00004 


0.00004 


0.00005 


0.00006 


AI2O3 (solid) 


0.00000 


0.00000 


0.00000 


0.00000 


0.00000 


AI2O3 (liquid) 


0.09425 


0.09378 


0.09343 


0.09293 


0.09178 


CO 


0.22434 


0.22374 


0.22328 


0.22259 


0.22085 


COCl 


0.00001 


0.00001 


0.00001 


0.00001 


0.00000 


CO2 


0.00785 


0.00790 


0.00793 


0.00799 


0.00810 


Cl 


0.00541 


0.00620 


0.00681 


0.00772 


0.01002 


CI2 


0.00001 


0.00001 


0.00001 


0.00001 


0.00001 


H 


0.02197 


0.02525 


0.02776 


0.03157 


0.04125 


HCl 


0.12021 


0.11900 


0.11808 


0.11668 


0.11321 


HCN 


0.00003 


0.00002 


0.00001 


0.00001 


0.00000 


HCO 


0.00003 


0.00002 


0.00002 


0.00002 


0.00001 


H2 


0.32599 


0.32380 


0.32215 


0.31968 


0.31362 


H2O 


0.08960 


0.08937 


0.08916 


0.08886 


0.08787 


NH2 


0.00001 


0.00001 


0.00001 


0.00000 


0.00000 


NH3 


0.00004 


0.00003 


0.00002 


0.00001 


0.00001 


NO 


0.00019 


0.00021 


0.00023 


0.00025 


0.00030 


N2 


0.09910 


0.09886 


0.09867 


0.09839 


0.09767 


0 


0.00010 


0.00014 


0.00016 


0.00021 


0.00036 


OH 


0.00262 


0.00297 


0.00324 


0.00364 


0.00458 


O2 


0.00001 


0.00001 


0.00002 


0.00002 


0.00004 



Source: From Ref. 5-13. 



TABLE 5-9. Calculated Variation of Thermodynamic Properties and Exit Gas Composition for an Aluminized Perchlorate Propellant with 
p] = 1500 psia and Various Exit Pressures at Shifting Equilibrium and Optimum Expansion 



X 

PQ 



o 

Z 















0 


0 


0 


0 


0 


0 


0 


0 




CN 










0 


0 


0 


0 


0 


0 


0 


0 


r- 


m 


CO 






0 


0 


0 


0 


0 


0 


0 


0 


0 


cs 


1-H 


CO 




0 


0 


0 


0 


0 


0 


0 


0 


0 


0 




0 


CO 




0 


os 


0 


0 


0 


0 


0 


0 


0 


0 


d 


d 


d 




d 


OS 


d 


d 


d 


d 


d 


d 


d 


d 










0 


CN 































0 


0 


0 


0 


0 


0 


0 


0 


CN 












0 


0 


0 


0 


0 


0 


0 


0 


»o 


so 


— H 






os 


0 


0 


0 


0 


0 


0 


0 


0 


»o 




— H 




0 


Os 


0 


0 


0 


0 


0 


0 


0 


0 


CN 


0 






0 


CO 


0 


0 


0 


0 


0 


0 


0 


0 


d 


d 




SO 


d 


OS 


d 


d 


d 


d 


d 


d 


d 


d 








<— 


0 


cs 
























SO 



































0 


0 


0 


0 


0 


0 


0 


0 














0 


0 


0 


0 


0 


0 


0 


0 


0 


CNJ 


(N 








0 


0 


0 


0 


0 


0 


0 


0 




lO) 






0 


OS 


0 


0 


0 


0 


0 


0 


0 


0 




0 


Os 


Os 


0 


00 


0 


0 


0 


0 


0 


0 


0 


0 


d 


d 


cn 


SO 


d 


OS 


d 


d 


d 


d 


d 


d 


d 


d 






CN 


0 


0 


CN 
























00 


CN 

































0 


00 


0 


0 


0 


0 


0 


0 




CN 










0 


0 


0 


0 


0 


0 


0 


0 


0 










OS 


0 


0 


0 


0 


0 


0 


0 


0 


0 


0 


os 




r- 


r- 


0 


0 


0 


0 


0 


0 


0 


0 


q 




CN 


l> 


q 


00 


0 


0 


0 


0 


0 


0 


0 


0 




d 




l> 


CN 


os 


d 


d 


d 


d 


d 


d 


d 


d 








0 


0 


CN 



















o 

(N 















0 




CN 


CN 




0 


0 


0 














0 




0 


0 


0 


0 


0 


0 


0 


SO 


1 -H 






cn 


0 


0 


0 


0 


0 


0 


0 


0 


0 


0 










0 


0 


0 


0 


0 


0 


0 


0 


q 


CnJ 




in 


q 


00 


0 


0 


0 


0 


0 


0 


0 


0 


CN 


d 


cn 


06 




OS 


d 


d 


d 


d 


d 


d 


d 


d 








CN 


» 0 ) 


cs 



















CN 















CO 

0 


00 


0 

CN 


CO 

(N 


in 

in 


so 


CO 


0 


0 




0 






CO 


0 


CN 




0 


0 


0 


0 


0 


so 


so 


0 




CO 


»o 


0 


0 


0 


0 


0 


0 


0 


0 


00 


O) 


q 


CO 






0 


0 


0 


0 


0 


0 


0 


0 


06 

»o 






l> 




OS 

cs 


d 


d 


d 


d 


d 


d 


d 


d 

















r- 






os 


so 


os 




CO 


Lh 














0 


in 


00 


(N 


00 


CM 


CM 


0 


0 




SO 






0 


CO 


0 






0 


0 


0 


0 


0 




r- 


in 






0 


0 


0 


0 


0 


0 


0 


0 


0 


0 


B 


q 


in 


(N 


os 


q 


CO 


0 


0 


0 


0 


0 


0 


0 


0 


cS 


CN 


d 


d 


SO 




OS 


d 


d 


d 


d 


d 


d 


d 


d 


ss 

U 


0 




A 


CO 




CN 






















CLh Oh 






(U 3 

is -2 

0 D 

N e- 

N C 

1 I 



<D 






1 -iH 

0 

.2 


!Z3 

0 


C/3 3 


fa 


0 

0 - 

r- 




rs 


fG 


u 


K 


K 

C'l 




Oh 


Gh ’0 




fa 

, 


U 


u 


U 


0 


0 


0 








C( 

A1 


< 


< 


< 


< 


< 


< 



188 



AI 2 O 3 (solid) 0.00000 0.00000 0.09955 0.09969 0.09974 0.09976 0.09976 

AI 2 O 3 (liquid) 0.09425 0.09608 0.00000 0.00000 0.00000 0.00000 0.00000 



CO 0.22434 0.22511 0.22553 0.22416 0.22008 0.21824 0.21671 

CO 2 0.00785 0.00787 0.00994 0.01126 0.01220 0.01548 0.01885 

Cl 0.00541 0.00441 0.00074 0.00028 0.00009 0.00002 0.00000 

H 0.02197 0.01722 0.00258 0.00095 0.00030 0.00007 0.00001 

HCl 0.12021 0.12505 0.13635 0.13707 0.13734 0.13743 0.13746 



CNl 

in 

d 



<N 

l> 

O 

d 



o in 

O 'O 

o o 
o o 

O -H 

d d 



o 

o 

o 

o 

o 

d 



o 

o 

o 

o 

o 

d 



00 




0 




0 


0 


00 


in 


0 


'O 


0 


0 


CNJ 


in 


0 


0 


0 


0 


in 


r- 


0 


0 


0 


0 


<n 


0 


0 




0 


0 


d 


d 


d 


d 


d 


d 





'O 


0 




0 


CN 




o^ 


0 


'O 


0 


0 


00 


r- 


0 


0 


0 


0 




r- 


0 


0 


0 


0 


<n 


0 


0 




0 


0 


d 


d 


d 


d 


d 


d 



0 


i> 


0 


00 


0 


in 


<n 




0 


in 


0 


0 


\o 


Os 


0 


0 


0 


0 




i> 


0 


0 


0 


0 




0 


0 




0 


0 


d 


d 


d 


d 


d 


d 



<n 






00 


0 


o^ 


0 


Os 


0 




0 


0 




0 


0 


0 


0 


0 




00 


0 


0 


0 


0 




0 


0 




0 


0 


d 


d 


d 


d 


d 


d 









0 


in 


CNJ 




0 


— H 


in 


0 


r-- 


0 


i> 


0 


G^ 


0 


1-^ 




00 


0 


On 


0 


0 


<n 


0 


0 


0 


0 


0 


d 


d 


d 


d 


d 


d 



o^ 


0 


o^ 


0 


0 


CNJ 


o^ 




— H 


— H 


— H 




in 


o^ 


0 


G^ 


0 


CM 


CN 


00 


0 


Os 


0 


0 


<n 


0 


0 


0 


0 


0 


d 


d 


d 


d 


d 


d 



n ^ O C4 K 
K Z Z O O 



189 



190 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



TABLE 5-10. Typical Gas Characteristics for Fuel-Rich Liquid Propellant Gas 
Generators 



Propellant 


Ti (K) 


k 


Gas Constant R 
(ft-lbf/lbm-°r) 


Oxidizer-to- 
Fuel Ratio 


Specific Heat 
Cp (kcal/kg-K) 


Liquid oxygen 


900 


1.370 


421 


0.919 


1.99 


and liquid 
hydrogen 


1050 


1.357 


375 


1.065 


1.85 




1200 


1.338 


347 


1.208 


1.78 


Liquid oxygen 


900 


1.101 


45.5 


0.322 


0.639 


and kerosene 


1050 


1.127 


55.3 


0.423 


0.654 




1200 


1.148 


64.0 


0.516 


0.662 


Nitrogen 


1050 


1.420 


87.8 


0.126 


0.386 


tetroxide and 

dimethyl 

hydrazine 


1200 


1.420 


99.9 


0.274 


0.434 



PROBLEMS 

1. Explain the physical or chemical reasons for a maximum value of specific impulse at 
a particular mixture ratio of oxidizer to fuel. 

2. Explain why, in Table 5-8, the relative proportion of monatomic hydrogen and mona- 
tonic oxygen changes markedly with different chamber pressures and exit pressures. 

3. This chapter contains several charts for the performance of liquid oxygen and RP-1 
hydrocarbon fuel. By mistake the next shipment of cryogenic oxidizer contains at least 
15% liquid nitrogen. Explain what general trends should be expected in the results 
of the next test in the performance values, the likely composition of the exhaust gas 
under chamber and nozzle conditions, and the optimum mixture ratio. 

4. A mixture of perfect gases consists of 3 kg of carbon monoxide and 1.5 kg of nitrogen 
at a pressure of 0.1 MPa and a temperature of 298.15 K. Using Table 5-1, find (a) the 
effective molecular mass of the mixture, (b) its gas constant, (c) specific heat ratio, 
(d) partial pressures, and (e) density. 

Answers: (a) 28 kg/kg-mol, (b) 297 J/kg-K, (c) 1.40, (d) 0.0666 and 0.0333 MPa, (e) 
1.13 kg/m^ 

5. Using information from Table 5-2, plot the value of the specific heat ratio for carbon 
monoxide (CO) as a function of temperature. Notice the trend of this curve; it is typical 
of the temperature behavior of other diatomic gases. 

Answers: k = 1.22, at 3500 K, 1.30 at 2000 K, 1.39 at 500 K. 

6. Modify and tabulate two entries in Table 5-5 for operation in the vacuum of space, 
namely oxygen/hydrogen and nitrogen tetroxide/hydrazine. Assume the data in the 
table represents the design condition. 



SYMBOLS 191 



7. Various experiments have been conducted with a liquid monopropellant called 
nitromethane (CH 3 NO 2 ), which can be decomposed into gaseous reaction products. 
Determine the values of T, 971, k, c*,Cp, and /j using the water-gas equilibrium 
conditions. Assume no dissociations and no O 2 . 

Answers: 2470 K, 20.3 kg/kg-mol, 1.25, 2394 m/sec, 1.57, 244 sec. 

8. The figures in this chapter show several parameters and gas compositions of liquid 
oxygen burning with RP-1, which is a kerosene-type material. For a mixture ratio of 
2.0, use the compositions to verify the molecular mass in the chamber and the specific 
impulse (frozen equilibrium flow in nozzle) in Fig. 5-1. 



SYMBOLS 

(Symbols referring to chemical elements, compounds, or mathematical operators 
are not included in this list.) 

a number of kilogram atoms 

A, throat area, m^ 

c* characteristic velocity, m/sec 

Cp specific heat per unit mass, J/kg-K 

Cp molar specific heat at constant pressure of gas mixture, J/kg-mol-K 

go acceleration of gravity at sea level, 9.8066 m/sec^ 

G Gibbs free energy for a propellant combustion gas mixture, J/kg 

AfG^ change in free energy of formation at 298.15 K and 1 bar 
Gj free energy for a particular species j, J/kg 

AH overall enthalpy change, J/kg or J/kg-mol 

AHj enthalpy change for a particular species j , J/kg 
ArH^ heat of reaction at reference 298.15 K and 1 bar, J/kg 
AfH^ heat of formation at reference 298.15 K and 1 bar, J/kg 
h enthalpy for a particular species, J/kg or J/kg-mol 

Is specific impulse, sec 

k specific heat ratio 

m number of gaseous species 

m mass flow rate, kg/sec 

9JI molecular mass (also called molecular weight) of gas mixture, kg/mol 

n total number of moles per unit mass (kg-mol/kg) of mixture 

tij moles of species j, kg-mol/kg 

P pressure of gas mixture, N/m^ 

R gas constant, J/kg-K 

R' universal gas constant, 8314.3 J/kg mol-K 

S entropy, J/kg mol-K 

T absolute temperature, K 

Tad adiabatic temperature, K 

U internal energy, J/kg-mol 



192 CHEMICAL ROCKET PROPELLANT PEREORMANCE ANALYSIS 



V gas velocity, m/sec 

V specific volume, m^/kg 

Xj mole fraction of species j 



Greek Letters 

p density, kg/m^ 



Subscripts 



a, b 
c, d 
i 



j 

mix 

ref 

1 

2 

3 



molar fractions of reactant species A or B 
molar fractions of product species C or D 
atomic species in a specific propellant 
constituents or species in reactants or products 
mixture of gases 

at reference condition (also superscript 0) 
chamber condition 
nozzle exit condition 
ambient atmospheric condition 



REFERENCES 

5-1. F. Van Zeggeren and S. H. Storey, The Computation of Chemical Equilibria, Cam- 
bridge University Press, Cambridge, 1970. 

5-2. S. S. Penner, Thermodynamics for Scientists and Engineers , Addison- Wesley Pub- 
lishing, Reading, MA, 1968. 

5-3. S. I. Sandler, Chemical and Engineering Thermodynamics , John Wiley & Sons, 
New York, 1999. 

5-4. R. H. Dittman and M. W. Zemansky Heat and Thermodynamics , 7th ed., McGraw- 
Hill Book Company, New York, 1996. 

5-5. K. Denbigh, The Principles of Chemical Equilibrium, 4th ed., Cambridge Univer- 
sity Press, Cambridge, 1981. 

5-6. K. K. Kuo, Principles of Combustion, 2nd ed., John Wiley & Sons, Hoboken, NJ, 
2005. 

5-7. JANAE Thermochemical Tables, Dow Chemical Company, Midland, Ml, Series A 
(June 1963) through Series E (January 1967). 

5-8. M. W. Chase, C. A. Davies, J. R. Downey, D. J. Frurip, R. A. McDonald, and A. 
N. Syverud, JANAE Thermochemical Tables, 3rd ed.. Part I, Journal of Physical 
and Chemical Reference Data, Vol. 14, Supplement 1, American Chemical Society, 
American Institute of Physics, and National Bureau of Standards, 1985. 



REFERENCES 193 



5-9. D. D. Wagman et al., “The NBS Tables of Chemical Thermodynamic Properties,” 
Journal of Physical and Chemical Reference Data, Vol. 11, Supplement 2, Amer- 
ican Chemical Society, American Institute of Physics, and National Bureau of 
Standards, 1982. 

5-10. J. B. Pedley, R. D. Naylor, and S. P. Kirby, Thermochemical Data of Organic 
Compounds, 2nd ed.. Chapman & Hall, London, 1986. 

5-11. B. J. McBride, S. Gordon, andM. Reno, “Thermodynamic Data for Fifty Reference 
Elements,” NASA Technical Paper 3287, January 1993. 

5-12. B. J. McBride and S. Gordon, “Computer Program for Calculating and Fitting 
Thermodynamic Functions,” NASA Reference Publication 1271 , November 1992. 

5-13. S. Gordon and B. J. McBride, “Computer Program for Calculation of Complex 
Chemical Equilibrium Compositions and Applications, Vol. 1: Analysis” (October 
1994) and “Vol. 2: User Manual and Program Description” (June 1996), NASA 
Reference Publication 1311 . 

5-14. S. Gordon and B. J. McBride, “Finite Area Combustor Theoretical Rocket Perfor- 
mance,” NASA TM 100785, April 1988. 



CHAPTER 6 



LIQUID PROPELLANT ROCKET 
ENGINE FUNDAMENTALS 



This chapter gives an overview of liquid propellant rocket engines. It is the first 
of six chapters devoted to this subject. It identifies the types of liquid rocket 
engines, their key components, different propellants, and tank conhgurations. 
It also discusses two types of feed systems, engine cycles, propellant tanks, 
their pressurization subsystems, engine controls, valves, piping, and structure. 
Chapter 7 covers liquid propellants in more detail. Chapter 8 describes thrust 
chambers (and their nozzles), small thrusters, and heat transfer. Chapter 9 is 
about the combustion process and Chapter 10 discusses turbopumps. Chapter 11 
discusses engine design, engine controls, propellant budgets, engine balance and 
calibration, and overall engine systems. 

In this book a liquid propellant rocket propulsion system consist of a rocket 
engine and a set of tanks for storing and supplying propellants. It has all the 
hardware components and the propellants necessary for its operation, that is, for 
producing thrust. See Refs. 6-1 and 6-2. They are described in this chapter. 
A rocket engine consists of one or more thrust chambers, a feed mechanism 
for feeding the propellants from their tanks into the thrust chamber(s), a power 
source to furnish the energy for the feed mechanism, suitable plumbing or piping 
to transfer the liquid propellants, a structure to transmit the thrust force, and 
control devices (including valves) to start and stop and sometimes also vary the 
propellant flow and thus the thrust. Propellants are either pushed out of their 
tanks by high-pressure gas or they are pumped by pumps to the thrust chambers. 
Figure 6-1 shows the Space Shuttle main engine, a large sophisticated high- 
performance liquid propellant rocket engine. The index lists the pages where 
information on the Space Shuttle, its main rocket engine, and various components 
can be found. 



194 




195 



FIGURE 6-1. Two views of the Space Shuttle Main Engine (SSME). Its flowsheet is in Figure 6-12 and some component data are in 
Chapter 10. (Courtesy of Pratt & Whitney Rocketdyne and NASA.) 



196 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



In some applications rocket engines may also include a thrust vector con- 
trol system (explained in Chapter 18), a random variable thrust feature (see 
Section 8.5), an engine condition monitoring or engine health monitoring sub- 
system (see Section 11.4), and various instrumentation/measuring devices (see 
Chapter 21). The tanks, which store the propellants, and the subsystem for pres- 
surizing the tanks with gas are in this chapter and are considered in this book to 
be part of the rocket propulsion system.* 

The design of any propulsion system is tailored to fit a specific mission require- 
ment as explained in Chapter 19. These requirements are usually stated in terms 
of the application, such as anti-aircraft missile or second stage of a space launch 
vehicle, flight velocity increment, flight path and flight maneuvers, launch sites, 
minimum life (in storage or in orbit), or number of vehicles to be delivered. The 
requirement usually include constraints of the inert engine weight, cost, or safety 
provisions. Other criteria, constraints and the selection process are in Chapter 19. 

From the mission requirements and definition one can derive the propulsion 
system and the engine requirements , which include the thrust-time prohle, the 
minimum specific impulse, number of thrust chambers, total impulse, number 
of restarts, if any, likely propellants, or constraints of engine masses or engine 
sizes. Some engine parameters, such as thrust, chamber pressure, mixture ratio, or 
nozzle exit area ratio, can be analytically optimized for a specific mission. Other 
engine parameters can be selected based on experience and/or design studies, 
including the feed system, the arrangement of the engine components, engine 
cycle, thrust modulation, or alternate methods of thrust vector control. Two or 
more preliminary or conceptual designs can be compared with the objective of 
arriving at a selected propulsion design for the mission. 

Tables 1-3, 11-2, and 11-3 give typical data for selected rocket engines. 
Many different types of rocket engines have been studied, built, and flown, rang- 
ing in thrust size from less than 0.01 Ibf to over 1.75 million pounds, with 
one-time operation or multiple starts (some have over 150,000 restarts), with or 
without thrust modulation (called throttling), single use or reusable, arranged as 
single engines, or in clusters of multiple units. 

One way to categorize liquid propellant rocket engines is described in 
Table 6-1. There are two categories, namely those used for boosting a 
payload and imparting a significant velocity increase to a payload and auxiliary 
propulsion for trajectory adjustments and attitude control. Liquid propellant 

*The responsibilities for the design, development, fabrication, and operation of a propulsion system 
are usually shared between a rocket engine organization and a flight vehicle organization. However, 
the allocations of the responsibilities for components or subsystems have not been rigid or consistent 
in the literature and in actual industry practice. For example, some vehicle design/development 
organizations have considered the tanks and parts of the engine structure to be really a part of their 
vehicle. The tank pressurization system has in various scenarios been considered to be part of either 
the engine, the propulsion system, or the vehicle. For some reaction control systems the vehicle 
developer often assumed the responsibility for the propulsion systems and obtained only the small 
thrusters and their small propellant valves from a rocket engine company. In some programs, such as 
the peacekeeper missile fourth stage, the rocket engine developer has developed not only the engine, 
but also much of the vehicle stage with its propellant tanks and pressurization system. 



LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 197 



TABLE 6-1. Characteristics of Two Categories of Liquid Propellant Rocket Engines 



Purpose 


Boost Propulsion 


Auxiliary Propulsion 


Mission 


Impart significant velocity to 


Attitude control, minor space 




propel a vehicle along its 


maneuvers, trajectory 




flight path 


corrections, orbit maintenance 


Applications 


Booster stage and upper stages 


Spacecraft, satellites, top stage 




of launch vehicles, large 


of antiballistic missile, space 




missiles 


rendezvous 


Total impulse 


High 


Low 


Number of thrust 


Usually 1; sometimes 4, 3, or 2 


Between 4 and 24 


chambers per 
engine 


Thrust level 


High; 4500 N up to 7,900,000 N 


Small; 0.001 up to 4500 N, a 




or 1000-1,770,000 Ibf 


few go up to 1000 Ibf 


Feed system. 


Mostly turbopump type; 


Pressurized feed system with 


typical 


occasionally pressurized feed 


high-pressure gas supply 


Tank pressure 


system for smaller thrusts 
0.138-0.379 MPa or 20-55 psi 


0.689-17.23 MPa or 100-2500 


range 




psi 


Most common 


Propellant cooled 


Radiation cooled 


cooling method 


Propellants (see 


Cryogenic and storable liquids 


Storable liquids, 


next section) 




monopropellants, and/or stored 


Chamber pressure 


2.4-21 MPa or 350-3600 psi 


cold gas 

0.14-2.1 MPa or 20-300 psi 


Number of starts 


Usually no restart; sometimes 


Several thousand starts are 


during a single 


one, but up to four in some 


typical for small thrusters 


mission 


cases 




Cumulative 


Up to a few minutes 


Up to several hours 


duration of 
firing 


Shortest firing 


Typically 5-40 sec 


0.02 sec typical for small 


duration 




thrusters 


Time elapsed to 


Up to several seconds 


Usually very fast, 0.004-0.080 


reach full thrust 




sec 


Life in space 


Hours, days, or months 


10 years or more in space 



rocket engine systems can be classified in several other ways. They can he 
reusable (like the Space Shuttle main engine or a rocket engine for quick ascent 
or maneuvers of fighter aircraft) or suitable for a single flight only (as the 
engines in other launch vehicles), and they can he restartable, like a reaction 
control engine, or single firing, as in space launch vehicle boosters. They can 
also be categorized by their propellants, application, or stage, such as an upper 
stage or booster stage, their thrust level, and hy the feed system type (pressurized 
or turbopump). 



198 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



The thrust chamber or thruster is the combustion device where the liquid pro- 
pellants are metered, injected, atomized, mixed, and burned to form hot gaseous 
reaction products, which in turn are accelerated and ejected at a high velocity 
to impart a thrust force. A thrust chamber has three major parts: an injector, 
a combustion chamber, and a nozzle. In a cooled thrust chamber, one of the 
propellants (usually the fuel) is circulated through cooling jackets or a special 
cooling passage to absorb the heat that is transferred from the hot reaction gases 
to the thrust chamber walls (see Figs. 8-2 and 8-9). A radiation-cooled thmst 
chamber uses a special high-temperature material, such as niobium metal, which 
can radiate away its excess heat. There are uncooled or heat-absorbing thmst 
chambers, such as those using ablative materials. Thrust chambers are discussed 
in Chapter 8. 

There are two types of feed systems used for liquid propellant rocket engines: 
those that use pumps for moving the propellants from their flight vehicle tanks to 
the thmst chamber and those that use high-pressure gas for expelling or displacing 
their propellants from their tanks. They are discussed further in Chapter 10 and 
in Section 6.3. 

Tables 19-1 to 19-4 compare the advantages and disadvantages of liquid 
propellant rocket engines and solid propellant rocket motors. 



6.1. TYPES OF PROPELLANTS 

The liquid propellants, which are the working substance of rocket engines, con- 
stitute the fluid that undergoes chemical and thermodynamic changes. The term 
liquid propellant embraces all the various liquids used and may be one of the 
following: 

1. Oxidizer (liquid oxygen, nitric acid, nitrogen tetroxide, etc.) 

2. Fuel (kerosene, alcohol, liquid hydrogen, etc.) 

3. Chemical compound or mixture of oxidizer and fuel ingredients, capable 
of self-decomposition, such as hydrazine 

4. Any of the above, but with a gelling agent. Used only rarely. 

All are described in Chapter 7. 

A bipropellant has two separate liquid propellants, an oxidizer and a fuel. They 
are the most common type. They are stored separately and are not mixed outside 
the combustion chamber. A hypergolic bipropellant combination is self-igniting 
upon contact between the oxidizer and the fuel. A nonhypergolic bipropellant 
combination needs energy to start its combustion (e.g., heat or electric discharge) 
and its engine needs an ignition system. 



6.1 . TYPES OF PROPELLANTS 1 99 



A monopropellant contains an oxidizing agent and combustible matter in a 
single substance. It may be a mixture of several compounds or it may be a 
homogeneous material, such as hydrogen peroxide or hydrazine. Monopropel- 
lants are stable at ordinary atmospheric conditions but decompose and yield hot 
combustion gases when heated or catalyzed. 

A cold gas propellant (e.g., gaseous nitrogen or air) is stored at ambient 
temperature but at very high pressure, gives a low performance, allows a simple 
system, and is usually very reliable. It has been used for roll control and attitude 
control. 

A cryogenic propellant is liquified gas at low temperature, such as liquid oxy- 
gen (— 183°C) or liquid hydrogen (— 253°C). Provisions for venting the storage 
tank and minimizing vaporization losses are necessary with this type. 

Storable propellants (e.g., nitric acid or gasoline) are liquid at ambient temper- 
ature and at modest pressure and can be stored for long periods in sealed tanks. 
Space- storable propellants are liquid in the environment of space; this storability 
depends on the specific tank design, thermal conditions, and tank pressure. An 
example is ammonia. 

A gelled propellant is a thixotropic liquid with a gelling additive. It behaves 
like a jelly or thick paint. It will not spill or leak readily, can flow under pressure, 
will burn, and is safer in some respects. It is described at the end of Chapter 7. 
Gelled propellants have been used in a few experimental rocket engines. 

A hybrid propellant usually has a liquid oxidizer and a solid fuel. It is dis- 
cussed in Chapter 16. 

The propellant mixture ratio for a bipropellant is the ratio at which the oxidizer 
and fuel are mixed and react to give hot gases. The mixture ratio r is defined as 
the ratio of the oxidizer mass flow rate m„ and the fuel mass flow rate htf or 

r = hiolhif (6-1) 

As explained in Chapter 5 the mixture ratio defines the composition and tem- 
perature of the combustion or reaction products. It is usually chosen to give a 
maximum value of specific impulse or Ti/OTl, where T\ is the absolute combus- 
tion temperature and Tl is the average molecular mass of the reaction gases (see 
Eq. 3-16 or Fig. 3-2). For a given thrust F and a given effective exhaust veloc- 
ity c, the total propellant flow is given by Eq. 2-6; namely, m = vv/go = F /c. 
The relationships between r, rh, tko, and thf are 



tho -\- thf = til 


(6-2) 


tho = rhijlr 1) 


(6-3) 


thf = m /(r -|- 1) 


(6-4) 



These same four equations are valid when w and vv (weight and weight flow 
rate) are substituted for m and hi. Calculated performance values for a number 



200 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



of different propellant combinations are given for specific mixture ratios in Table 
5-5. Physical properties and a discussion of several common liquid propellants 
and their safety concerns are described in Chapter 7. 

Example 6-1. A liquid oxygen-liquid hydrogen rocket thrust chamber of 10,000-lbf 
thrust operates at a chamber pressure of 1000 psia, a mixture ratio of 3.40, has exhaust 
products with a mean molecular mass of 8.90 Ibm/lb-mol, a combustion temperature of 
4380°F, and a specific heat ratio of 1.26. Determine the nozzle throat area, nozzle exit area 
for optimum operation at an altitude where = p 2 = 1.58 psia, the propellant weight 
and volume flow rates, and the total propellant requirements for 2.5 min of operation. 
Assume that the actual specific impulse is 97% of the theoretical value and that the thrust 
coefficient is 98% of ideal value for this problem. 

SOLUTION. The exhaust velocity for an optimum nozzle is determined from Eq. 3-16, 
but with a correction factor of go for the foot-pound system: 



V2 



2gpk R'Ti 

k-l m 







2 X 32.2 X 1.26 1544 x 4840 



0.26 



8.9 



(1 - 0 . 001580 - 205 ) 



13,890 ft/sec 



The theoretical specific impulse is c/go, or in this optimum expansion case V 2 /go or 
13,890/32.2 = 431 sec. The actual specific impulse is 0.97 x 431 = 418 sec. The theo- 
retical or ideal thrust coefficient can be found from Eq. 3-30 or from Fig. 3-6 (p 2 = pa) 
and for a pressure ratio pi/p 2 = 633 to be Cf = 1.76. The actual thrust coefficient is 
slightly less, say 98% or Cf = 1.72. The throat area required is found from Eq. 3-31. 

A, = F/iCfPx) = 10,000/(1.72 x 1000) = 5.80 in.^ (2.71 in. diameter) 

The optimum area ratio can be found from Eq. 3-25 or Eig. 3-5 to be 42. The exit area 
is 5.80 X 42 = 244 in.^ (17.6 in. diameter). The weight density of oxygen is 71.1 Ibf/ft^ 
and of hydrogen is 4.4 Ibf/ft^. The propellant weight flow rates (Eqs. 2-5, 6-3, and 6-4) 
are 



w = F/E = 10,000/418 = 24.0 Ibf/sec 

= wr/{r -H 1) = 24.0 x 3.40/4.40 = 18.55 Ibf/sec 
Wf = w/{r -H 1) = 24/4.40 = 5.45 Ibf/sec 

The volume flow rates are determined from the densities and the weight flow rates: 

V„ = Wo/Po = 18.55/71.1 = 0.261 ftVsec 
Vf = Wf/pf = 5.45/4.4 = 1.24 ftVsec 



6.2. PROPELLANT TANKS 201 



For 150 sec of operations (arbitrarily allow the equivalent of two additional seconds 
for start and stop transients and unavailable residual propellant), the weight and volume 
of required propellant are 

Wo = 18.55 X 152 = 2820 Ibf of oxygen 
w/ = 5.45 X 152 = 828 Ibf of hydrogen 
Vo = 0.261 X 152 = 39.7 ft^ of oxygen 
Vf = 1.24 X 152 = 188.5 ft^ of hydrogen 

Note that, with the low-density fuel, the volume flow rate and therefore the tank volume 
of hydrogen are 5 times as large compared to the oxidizer. Hydrogen is unique and in its 
liquid state has a very low density. 



6.2. PROPELLANT TANKS 

In liquid bipropellant rocket engine systems the propellants are usually stored in 
an oxidizer tank and a fuel tank within the flying vehicle. Monopropellant rocket 
engine systems have, of course, only one propellant tank. There are usually 
also one or more high-pressure gas tanks, the gas being used to pressurize the 
propellant tanks. However, as will be discussed in Section 6.5, there are tank 
pressurization schemes using heated gas from the engine without the need for 
a high-pressure heavy gas storage tank. Tanks can be arranged in a variety of 
ways, and the tank design, shape, and location can be used to exercise some 
control over the change in the location of the vehicle’s center of gravity. Typical 
arrangements are shown in Fig. 6-2. Because the propellant tank has to fly, 
its mass is at a premium and the tank material is therefore highly stressed. 
Common tank materials are aluminum, stainless steel, titanium, alloy steel, and 
fiber-reinforced plastics with an impervious thin inner liner of metal to prevent 
leakage through the pores of the fiber-reinforced waffs. Chapter 8 of Ref. 6-f 
describes the design of propellant tanks. 

The extra volume of gas above the propellant in sealed tanks is called ullage. 
It is necessary space that allows for thermal expansion of the propellant liquids, 
for the accumulation of gases that were originally dissolved in the propellant, 
or for gaseous products from slow reactions within the propellant during stor- 
age. Depending on the storage temperature range, the propellants’ coefficient of 
thermal expansion, and the particular application, the ullage volume is usually 
between 3 and 10% of the tank volume. Once propellant is loaded into a tank, 
the ullage volume (and, if it is sealed, also its pressure) will change as the bulk 
temperature of the propellant varies. 

The expulsion efficiency of a tank and/or propellant piping system is the 
amount of propellant expelled or available for propulsion divided by the total 
amount of propellant initially present. Typical values are 97 to 99.7%. The losses 



202 LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 



A 



i\ 




Spherical tanks 




Tandem tanks, 
external piping 




Tandem tanks 
with common 
bulkhead, 
internal piping 



3 Oxidizer 




FIGURE 6-2. Simplified sketches of typical tank arrangements for large turbopump-fed 
liquid propellant rocket engines. 



are unavailable propellants that are trapped in grooves or corners of pipes, fit- 
tings, and valves, are wetting the walls, retained by surface tension, or caught 
in instrument taps. This residual propellant is not available for combustion and 
must be treated as inert mass, causing the vehicle mass ratio to decrease slightly. 
In the design of tanks and piping systems, an effort is made to minimize the 
residual propellant. 

The optimum shape of a propellant tank (and also a gas pressurizing tank) is 
spherical because for a given volume it results in a tank with the least weight. 
Small spherical tanks are often used with reaction control engine systems, where 
they can be packaged with other vehicle equipment. Unfortunately, the larger 
spheres, which are needed for the principal propulsion systems, are not very 
efficient for using the space in a flight vehicle. These larger tanks are often made 
integral with the vehicle fuselage or wing. Most are cylindrical with half ellipses 
at the ends, but they can be irregular in shape. A more detailed discussion of 
tank pressurization is given in Section 6.5. 

Cryogenic propellants cool the tank wall temperature far below the ambient 
air temperature. This causes condensation of moisture from the air on the outside 
of the tank and usually also formation of ice during the period prior to launch. 
The ice is undesirable because it increases the vehicle inert mass. Also, as pieces 



6.2. PROPELLANT TANKS 203 



of ice are shaken off or break off during the initial flight, these pieces can damage 
the vehicle; for example, the ice from the Shuttle’s cryogenic tank can hit the 
orbiter vehicle. 

For an extended storage period, cryogenic tanks are usually thermally insu- 
lated; porous external insulation layers have to be sealed to prevent moisture from 
being condensed inside the insulation layer. With liquid hydrogen it is possible 
to liquify or solidify the ambient air on the outside of the fuel tank. Even with 
heavy insulation and low-conductivity structural tank supports, it is not possible 
to prevent the continuous evaporation of the cryogenic fluid. Even with good 
thermal insulation, all cryogenic propellants evaporate slowly during storage and 
therefore cannot be kept in a vehicle for more than perhaps a week without 
rehlling of the tanks. Eor vehicles that need to be stored or to operate for longer 
periods, a storable propellant combination is preferred. 

Prior to loading very cold cryogenic propellant into a flight tank, it is necessary 
to remove or evacuate the air to avoid forming solid air particles or condensing 
any moisture as ice. These frozen particles would plug up injection holes, cause 
valves to freeze shut, or prevent valves from being fully closed. Tanks, piping, and 
valves need to be chilled or cooled down before they can contain cryogenic liquid 
without excessive bubbling. This is usually done by letting an initial amount of 
cryogenic liquid absorb the heat from the relatively warm hardware prior to 
engine start. This initial cool-down propellant is vaporized and vented through 
appropriate vent valves and cannot be used for propulsion. 

If the tank or any segment of piping containing low-temperature cryogenic 
liquid is sealed for an extended period of time, heat from ambient-temperature 
hardware will result in evaporation, and this will greatly raise the pressure 
until it exceeds the strength of the container (see Ref. 6-3). Controlled self- 
pressurization can be difficult to achieve. Uncontrolled self-pressurization will 
cause a failure, usually a major leak or even a tank explosion. All cryogenic 
tanks and piping systems are therefore vented during storage on the launch pad, 
equipped with pressure safety devices (such as burst diaphragms or relief valves), 
and the evaporated propellant is allowed to escape from its container. Eor long- 
term storage of cryogenic propellants in space vacuum (or on the ground) some 
form of a powered refrigeration system is needed to recondense the vapors and 
minimize evaporation losses. The tanks are usually rehlled or topped off just 
before launch to replace the evaporated and vented cool-down propellant. When 
the tank is pressurized, just before launch, the boiling point is usually raised 
slightly and the cryogenic liquid can usually absorb the heat transferred to it 
during the several minutes of rocket firing. 

There are several categories of tanks in liquid propellant propulsion systems. 
There are a few exceptions to the pressure values listed below. 

1 . Eor pressurized feed systems the propellant tanks typically operate at an 
average pressure between 1.3 and 9 MPa or about 200 to 1800 Ibf/in.^ 
These tanks have thick walls and are heavy. 



204 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 

2. For high-pressure gas (used to expel the propellants) the tank pressures are 
much higher, typically between 6.9 and 69 MPa or 1000 to 10,000 Ibf/in.^. 
These tanks are usually spherical for minimum inert mass. Several small 
spherical tanks can be connected together, and then they are relatively easy 
to place within the confined space of a vehicle. 

3. For turbopump feed systems it is necessary to pressurize the propellant 
tanks slightly (to suppress pump cavitation as explained in Sections 10.3 
and 10.4) to average values of between 0.07 and 0.34 MPa or 10 to 50 
Ibf/in^. These low pressures allow thin tank walls, and therefore turbopump 
feed systems have relatively low inert tank weights. 

Liquid propellant tanks can be difficult to empty under side accelerations, 
zero-g, or negative-g conditions during flight. Special devices and special types 
of tanks are needed to operate under these conditions. Some of the effects that 
have to be overcome are described below. 

The oscillations and side accelerations of vehicles in flight can cause sloshing 
of the liquid in the tank, very similar to a glass of water that is being jiggled. In 
an anti-aircraft missile, for example, the side accelerations can be large and can 
initiate severe sloshing. Typical analysis of sloshing can be found in Refs. 6-4 
and 6-5. When the tank is partly empty, sloshing can uncover the tank outlet 
and allow gas bubbles to enter into the propellant discharge line. These bubbles 
can cause major combustion problems in the thrust chambers; the aspirating 
of bubbles or the uncovering of tank outlets by liquids therefore needs to be 
avoided. Sloshing also causes shifts in the vehicle’s center of gravity and makes 
flight control difficult. 

Vortexing can also allow gas to enter the tank outlet pipe; this phenomenon 
is similar to the Coriolis force effects in bath tubs being emptied and can be 
augmented if the vehicle spins or rotates in fight. Typically, a series of internal 
baffles is often used to reduce the magnitude of sloshing and vortexing in tanks 
with modest side accelerations. A positive expulsion mechanism can prevent gas 
from entering the propellant piping under multidirectional major accelerations 
or spinning (centrifugal) acceleration. Both the vortexing and sloshing can also 
greatly increase the unavailable or residual propellant, and thus cause a reduction 
in vehicle performance. 

In the gravity-free environment of space, the stored liquid will float around 
in a partly emptied tank and may not always cover the tank outlet, thus allow- 
ing gas to enter the tank outlet or discharge pipe. Figure 6-3 shows that gas 
bubbles have no orientation. Various devices have been developed to solve this 
problem: namely, positive expulsion devices and surface tension devices. The 
positive expulsion tank design includes movable pistons, inflatable flexible blad- 
ders, or thin movable, flexible metal diaphragms. Surface tension devices rely on 
surface tension forces to keep the outlet covered with liquid. 

Several basic types of positive expulsion devices have been used successfully 
in propellant tanks of pressurized feed systems. They are compared in Table 6-2 



6.2. PROPELLANT TANKS 205 



Liquid out 



Outlet 





tension propellant 
management device 
showing one particular 
liquid distribution during 
upward acceleration 



in a zero-gravity environment; surface 
liquid. 



FIGURE 6-3. Ullage bubbles can float around 
tension device can keep tank outlet covered with 



and shown in Fig. 6-4 for simple tanks. These devices mechanically separate 
the pressurizing gas from the liquid propellant in the propellant tank. Separation 
is useful for these reasons: 

1 . It prevents pressurizing gas from dissolving in the propellant and propellant 
from evaporating. Dissolved pressurizing gas dilutes the propellant, reduces 
its density as well as its specific impulse, and makes the pressurization 
inefficient. 

2. It allows moderately hot and reactive gases (such as gas generated by gas 
generators) to be used for pressurization, and this permits a reduction in 
pressurizing system mass and volume. The mechanical separation prevents 
a chemical reaction between the hot gas and the propellant, prevents gas 
from being dissolved in the propellant, and reduces the heat transfer to the 
liquid. 

3. In some cases tanks containing toxic propellant must be vented without 
spilling any toxic liquid propellant or its vapor. For example, in servicing 
a reusable rocket, the tank pressure needs to be relieved without venting 
or spilling potentially hazardous material. 



TABLE 6-2. Comparison of Propellant Expulsion Methods for Spacecraft Hydrazine Tanks 



<D C ^ 

^ .2 S 

c h 

c/5 H 



D 

Oh 

X 

W 





E 


00 


00 


.s 


c3 


X 


X 


o 


X 


CP 


03 




s 



° ^ 
° s 

tc D 
£ CQ 



— ^ 
- .2 Si'S 
-S S ^ G. 

^ Oj Oh ^ 

£ s s 1 



I D 

K 



== .2 
^ P QJ 

D S "O 
X P ^ 



izi --r 
^ « 
w 



■y u 

j= 






S H 

I 

0J5 P ^ 

C O -p Oh 

_ .4-1 D . cn 

^ J S E 

cq Q u 



U 



X 

PJ 



T3 

O 

o 



T3 

O 



X 

X 



O 2 25 

> 73 

IZ2 D 

g 8 'S 

^ c > 

tf 



T3 



C 

O 

X 

tf 



X 

PJ 



1-1 

o 



Oh 

Oh 

< 



d a j 



T3 

o a 

too a 
>» •B 

’ o 






t 3 ;2 
^ o “ 

o S 



OJ > 

P X o 

-H W J 



■T^ ^ 

" c 

s .§ 

■y '3 o 

s S :s 

S § S 



O 

O 

cu 



T3 

O 

O 

a 



D 

13 xi 
^ ^ a 

-H W ffi 



X 

M 



T3 

O 

O 

a 



o 

o 

Oh 



> IZ3 

^ O 

o o 



^ -2 
o ^ 



o a a 
X o .a 
PJ z J 



c 

o 



03 

Vh > 

^ 2 • 
-^-T g bJ5 

■3 .S c 



~o 

o 



X, ■■^ ,i=> CL) "O 

^ 2 13 =a o 

s o T3 C3 .G 



9 « ^ 
O J ^ 



u 



t3 __ 

8 tS 3 _ 

60 4= '3 _3 
>4-3 G C6 

Lh Cd -4-* <H-< 

X D O 

> J Oh 



J3 'C Cfl 
w CJ (U (L) 



S 

O > 

.2 S 

CO JC 



° c 

c/3 ^ 



c ^ X 

^ 2 Oh i£, ^ 

"u G3 S u 

^ ^ -O cj 

w j S 



o 3 

S 60 
S 'S 

60 S 



X 

o 

o 




X 


c 

ID 


c/3 

D 


73 

D 


a 

i-i 

O 

*c 


’c 

o 


2 


'C 

D 

s 


00 

b 


o 


2 

e 


13 

D 

X 


o3 

D 


s 

D 

X 


2 

D 

X 


c/3 

X 


M-h 


O 

D 

00 


> 


-1 


j 


PJ 




U 











C ^ 



o3 



W J u 



*C/3 



00 



D 

s 



o >, 



D 8 
2 -O 
> o 

H-. 

w t- 



c 

o 

u 



a 
> 

OO g ID !« 

j ( i : S 



206 



6.2. PROPELLANT TANKS 207 




Sliding piston with seals 




Propellant 
outlet pipe 



FIGURE 6-4. Three concepts of propellant tanks with positive expulsion: (a) inflatable 
dual bladder; (b) rolling, peeling diaphragm; (c) sliding piston. As the propellant volume 
expands or contracts with changes in ambient temperature, the piston or diaphragm will 
also move slightly and the ullage volume will change during storage. 



A piston expulsion device permits the center of gravity (CG) to be accurately 
controlled and its location to be known during the engine operation. This is 
important in rockets with high side accelerations such as anti-aircraft missiles or 
space defense missiles, where the thrust vector needs to go through the vehicle’s 
CG; if the CG is not well known, unpredictable turning moments may be imposed 
on the vehicle. A piston also prevents sloshing or vortexing. 

Surface tension devices use capillary attraction for supplying liquid propellant 
to the tank outlet pipe. These devices (see Fig. 6-3) are often made of very 
fine (300 mesh) stainless steel wire woven into a screen and formed into tunnels 
or other shapes (see Refs. 6-6 and 6-7). These screens are located near the 
tank outlet and, in some tanks, the tubular galleries are designed to connect 
various parts of the tank volume to the outlet pipe sump. These devices work 
best in a relatively low-acceleration environment, when surface tension forces 
can overcome the inertia forces. 

The combination of surface tension screens, baffles, sumps, and traps is called 
a propellant management device. Although not shown in any detail, they are 
included inside the propellant tanks of Figs. 6-3 and 6-13. 



208 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 

High forces can be imposed on the tanks and thus on the vehicle by strong 
sloshing motions of the liquid and also by sudden changes in position of liquid 
mass in a partly empty tank during a gravity-free flight when suddenly accelerated 
by a relatively large thrust. These forces can be large and can cause tank failure. 
The forces will depend on the tank geometry, baffles, ullage volume, and its 
initial location and the acceleration magnitude and direction. 



6.3. PROPELLANT FEED SYSTEMS 

The propellant feed system has two principal functions; to raise the pressure of 
the propellants and to feed them at predicted mass flow rates to one or more thrust 
chambers. The energy for these functions comes either from a high-pressure gas, 
centrifugal pumps, or a combination of the two. The selection of a particular feed 
system and its components is governed primarily by the application of the rocket, 
the requirements mentioned at the beginning of this chapter, duration, number or 
type of thrust chambers, past experience, mission, and by general requirements 
of simplicity of design, ease of manufacture, low cost, and minimum inert mass. 
A classification of several of the more important types of feed system is shown 
in Fig. 6-5, and some are discussed in more detail in other parts of this book. All 
feed systems have piping, a series of valves, provisions for filling and usually also 



Liquid propellant 
feed systems 



Pressurized systems 



Turbopump systems 



Direct gas 
pressurization 



Flexible bag 
within tank 



Piston 

pressurization 



Pump 



By stored 




By vaporized 




By chemically 


inert gas 




propellant 




generated gas 



Multi-stage 




Sinale staae 


impellers 




impeller 



T 



As received 




Heated 


1 


1 


1 




1 


Regulated 




Blow down 


pressure 







Small portion of one 
propellant flow raised to 
precombustor pressure 
with additional impeller 



One pump 
each for fuel 
and oxidizer 



Two main 
pumps plus two 
booster pumps 




Gas power supply 
and gas discharge 



Precombustion 

chamber 



Separate gas 
generator 



Warm 
hydrogen 
from cooling 
jacket 



X 



Driven by high pressure 
liquid propellant (for 
booster pumps only) 



Hot gas bleed 
from main 
combustion 
chamber 



Flow through 
turbines in 
series 



Flow through 
turbines in 
parallel 



One turbine drives 
both fuel and 
oxidizer pumps 




Two turbines: one 
for fuel pump, one 
for oxidizer pump 




Four turbines; two 
for main pumps, two 
for booster pumps 


1 


^ 


J 



Exhaust overboard 




Exhaust feeds 




Exhaust into 


with low area 




into diverging 




injector of 


ratio nozzle 




nozzle section 




main combustor 



With gear 




Direct drive 


transmission 







FIGURE 6-5. Design options of feed systems for liquid propellant rocket engines. The 
more common types are designated with a double line at the bottom of the box. 



6.3. PROPELLANT FEED SYSTEMS 209 



for removing (draining and flushing) the liquid propellants, and control devices 
to initiate, stop, and regulate their flow and operation. See Ref. 6-1. 

In general, a gas pressure feed system gives a vehicle performance superior 
to a turbopump system when the total impulse or the mass of propellant is 
relatively low, the chamber pressure is low, the engine thrust-to-weight ratio is 
low (usually less than 0.6), and when there are repeated short-duration thrust 
pulses; the heavy- walled tanks for the propellant and the pressurizing gas usually 
constitute the major inert mass of the engine system. In a turbopump feed system 
the propellant tank pressures are much lower (by a factor of 10 to 40) and thus the 
vehicle’s tank masses are much lower (again by a factor of 10 to 40). Turbopump 
systems usually give a superior vehicle performance when the total impulse is 
large (higher Am) and the chamber pressure is higher. 

Local Pressures and Flows 

Key parameters of any feed system and any liquid propellant rocket engine are 
the flow magnitudes and the local pressures of the pressurizing gas subsystem, 
the oxidizer, and the fuel flow subsystem including the flow passages of thrust 
chambers. An inspection of the flow diagram of a relatively simple rocket engine 
with a pressurized feed system, similar to Fig. 1-3, shows that the gas flow splits 
into two branches, and the propellant flow splits into pipes leading to each of the 
thrust chambers. The highest pressure is in the high-pressure gas supply tank. The 
pressure drops in the pressurizing gas subsystem (pipes, valves, regulator) and 
then drops further in the liquid propellant flow subsystems (more pipes, valves, 
filters, injector, or cooling jacket) and the flows go into the thrust chamber, where 
the propellants burn at chamber pressure. The pressure reaches a minimum at the 
nozzle exit. If enough information is known about the geometric details and 
functions of the components and their flow passages, then it has been possible 
to use computer programs to obtain these pressure drops and flow distributions. 
If this analysis can be validated by data from previous pertinent tests, it will be 
more useful. Many rocket engine design organizations have developed their own 
computer programs for estimating pressures and flows in different parts of an 
engine. 

Analyzing and knowing the local branch flows and local pressures is important 
for the following reasons: 

1 . They are used in the stress analysis and sometimes in the thermal analysis 
of components and subsystems. 

2. They are needed in the calibration of the rocket engine, so that this engine 
will operate at the intended mixture ratio, chamber pressure, or thrust. 
This is accomplished by using control devices or simple orifices to adjust 
the pressures. This calibration also requires the proper balance of flows 
and pressures. For a feed system with one or more turbopumps, such as 
the one in Fig. 1-4, one needs to also include the rise in pressure as 
propellants flow through a pump. Furthermore there is another combustion 



210 



LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



device (gas generator or prebumer) or alternatively the evaporation of a 
cryogenic propellant in a cooling jacket, and this makes the feed system 
with turbopumps more complex. A more detailed discussion of engine 
calibration is given in Section 11.5. 

3. Measuring the actual flows and several key local pressures during engine 
ground tests or actual flight operation and then comparing these actual 
values with the calculated or predicted values makes is possible to iden- 
tify discrepancies between practice and theory. Such discrepancies give 
clues to possible malfunction, inadequate design, or poor fabrication, which 
can then be identified and probably corrected. If the real measured values 
can be compared with the analysis values in real time, some experimental 
test hardware can be saved from self-destruction. This can be the basis 
for a real-time engine health monitoring system, which is discussed in 
Section 11.4. 

Some of the analyses are aimed at transient conditions, such as during the start 
period, the shutdown period, or the change in thrust value (throttling). These tran- 
sient analyses provide for the filling of empty propellant passages with propellant, 
for different propellant temperatures, water hammer, valve reaction time, and the 
like. 



6.4. GAS PRESSURE FEED SYSTEMS 

One of the simplest and most common means of pressurizing the liquid pro- 
pellants is to force them out of their respective tanks by displacing them with 
high-pressure gas. Rocket engines with pressurized gas feed systems can be very 
reliable. References 6-1, 6-8, and 6-9 give additional information. A rocket 
engine with a gas-pressurized feed system was the first to be tested and flown 
(1926). Since then there have been two common types of pressurized feed sys- 
tems and both are still used often today. The first uses a gas pressure regulator 
in the gas feed line and its engine operates at essentially constant tank pressure 
and nearly constant thrust. 

It is shown schematically in Fig. 1-3 and consists of a high-pressure gas tank, 
a gas starting valve, a pressure regulator, propellant tanks, propellant valves, 
and feed lines. Additional components, such as filling and draining provisions, 
check valves, filters, flexible elastic bladders for separating the liquid from the 
pressurizing gas, and pressure sensors or gauges, are also often incorporated. 
After all tanks are filled, the high-pressure gas valve in Fig. 1-3 is remotely 
actuated and admits gas through the pressure regulator at a constant pressure to 
the propellant tanks. The check valves prevent mixing of the oxidizer with the 
fuel, particularly when the unit is not in an upright position. The propellants are 
fed to the thrust chamber by opening appropriate valves. When the propellants 
are completely consumed, the pressurizing gas can also scavenge and clean lines 
and valves of much of the liquid propellant residue. The variations in this system, 
such as the combination of several valves into one or the elimination and addition 



6.4. GAS PRESSURE EEED SYSTEMS 21 1 



of certain components, depend to a large extent on the application. If a unit is to 
be used and flown repeatedly, such as a space-maneuver rocket, it will include 
several additional features such as, possibly, a thrust-regulating device and a tank 
level gauge; they will not be found in an expendable, single-shot unit, which may 
not even have a tank-drainage provision. 

The second common type of gas pressure feed system is called a blow-down 
feed system. It is shown in Fig. 6-6 and discussed in Ref. 6-10. Here the 
propellant tanks are enlarged because they store not only the propellants but also 
the pressurizing gas at the initial maximum propellant tank pressure. There is no 
separate high-pressure gas tank and no pressure regulator. The expansion of the 
gas already in the tanks provides for the expulsion of the propellants from their 
tanks. The blow-down system can be lighter than a regulated pressure system, but 
the thrust and the pressures decrease steadily as the propellants are consumed. A 
comparison of these two common types is shown in Table 6-3. 

Different bipropellant pressurization concepts are evaluated in Refs. 6-1, 6-8, 
6-9, and 6-10. Table 6-4 lists various optional features aimed at satisfying 
particular design goals. Many of these features also apply to pump-fed systems, 
which are discussed in Section 6.6. Many feed systems or engines have some, 
but certainly not all, of the features listed in Table 6-4. With monopropellants 
the gas pressure feed system becomes simpler since there is only one propellant 
and not two, reducing the number of pipes, valves, and tanks. 




FIGURE 6-6. Simplified schematic diagram of a bipropellant blow-down pressurized 
gas feed system with two thrusters. 



212 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



TABLE 6-3. Comparison of Two Types of Gas Pressurization Systems 



Type 


Regulated Pressure 


Blowdown 


Pressure/thrust 


Stays essentially constant 


Decreases as propellant is consumed 


Gas storage 


In separate high-pressure tanks 


Gas is stored inside propellant tank 
with large ullage volume 
(30-60%) 


Required 

components 


Needs regulator, filter, gas valve, 
and gas tank 


Larger, heavier propellant tanks 


Advantages 


Constant-pressure feed gives Simpler system. Less gas required 

essentially constant propellant Can be less inert mass 

flow and approximately constant 
thrust, constant R and r 
Better control of mixture ratio 


Disadvantages 


Slightly more complex 
Regulator introduces a small 
pressure drop 

Gas stored under high pressure 
Shorter burning time 


Thrust decreases with burn duration 
Somewhat higher residual 
propellant due to less accurate 
mixture ratio control 
Thruster must operate and be stable 
over wide range of thrust values 
and modest range of mixture ratio 
Propellants stored under pressure; 
slightly lower R toward end of 
burning time 



An example of a complex man-rated pressurized feed system, the combined 
Space Shuttle Orbital Maneuver System (OMS) and the Reaction Control Sys- 
tem (RCS), is described in Figs. 6-7 and 6-8, and Ref. 6-11. There are three 
locations on the Space Shuttle for the RCS, as shown in Fig. 1-14: a forward 
pod and a right and left aft pod. Figures 6-7 and 6-8 and Table 6-5 refer to 
one of the aft pods only and show a combined OMS and RCS arrangement. 
The OMS provides thrust for orbit insertion, orbit circularization, orbit transfer, 
rendezvous, deorbit, and abort. The RCS provides thrust for attitude control (in 
pitch, yaw, and roll) and for small-vehicle velocity corrections or changes in 
almost any direction (translation maneuvers), such as are needed for rendezvous 
and docking; it can operate simultaneously with or separate from the OMS. 

The systems feature various redundancies, an automatic RCS thruster selec- 
tion system, various safety devices, automatic controls, sensors to allow a display 
to the shuttle’s crew of the system’s status and health, and manual command 
overrides. The reliability requirements are severe. Several key components, such 
as all the helium pressure regulators, propellant tanks, some valves, and about 
half the thrusters are duplicated and redundant; if one fails, another can still 
complete the mission. It is possible to feed up to 1000 Ibm of the liquid from 
the large OMS propellant tanks to the small RCS ones, in case it is neces- 
sary to run one or more of the small reaction control thrusters for a longer 



6.4. GAS PRESSURE FEED SYSTEMS 213 



TABLE 6-4. Typical Features of Liquid Propellant Feed Systems 

Enhance Safety 

Check valves to prevent backflow of propellant into the gas tank and inadvertent mixing of 
propllants inside flow passages 

Pressurizing gas should be inert, clean, and insoluble in propellant 

Burst diaphragms or isolation valves to isolate the propellants in their tanks and positively prevent 
leakage into the thrust chamber or into the other propellant tank during storage 
Isolation valves to shut off a section of a system that has a leak or malfunction 
Sniff devices to detect leak of hazardous vapor; used on some versions of Space Shuttle Orbiter 
Features that prevent an unsafe condition to occur or persist and shut down engine safely, such as 
relief valves or relief burst diaphragms to prevent tank overpressurization), or a vibration 
monitor to shut off operation in the case of combustion instability 

Provide Control 

Valves to control pressurization and flow to the thrust chambers (start/stop/throttle) 

Sensors to measure temperatures, pressures, valve positions, thrust, etc., and computers to 
monitor/analyze system status. Compare measured values with analytical estimates, issue 
command signals, and comect if sensed condition is outside predetermined limits 
Manned vehicle can require system status display and command signal override 
Fault detection, identification, and automatic remedy, such as shut-off isolation valves in 
compartment in case of fire, leak, or disabled thruster 
Control thrust (throttle valve) to fit a desired thrust-time flight profile 

Enhance Reliability 

Fewest practical number of components/subassemblies 

Filters to catch dirt in propellant lines, which could prevent valve from closing or small injector 
holes from being plugged up or bearings from galling 
Duplication of key components, such as redundant small thrusters, regulators, or check valves. If 
malfunction is sensed, then remedial action is to use the redundant spare component 
Heaters to prevent freezing of moisture or low-melting-point propellant 

Long storage life — use propellants with little or no chemical deterioration and no reaction with 
wall materials 



Provide for Reusability 

Provisions to drain propellants or pressurants remaining after operation 

Provision for cleaning, purging, flushing, and drying the feed system and refilling propellants and 
pressurizing gas 

Devices to check functioning of key components prior to next operation 
Features to allow checking of engine calibration and leak testing after operation 
Access for inspection devices for visual inspection of internal surfaces or components for damage 
or failure 



Enable Effective Propellant Utilization 

High tank expulsion efficiency with minimum residual, unavailable propellant 
Lowest possible ambient temperature variation and/or matched propellant property variation with 
temperature so as to minimize mixture ratio change and residual propellant 
Alternatively, measure remaining propellant in tanks (using a special gauge) and automatically 
adjust mixture ratio (throttling) to minimize residual propellant 
Minimize pockets in the piping and valves that cannot be readily drained or flushed 



214 LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 



RCS helium RCS propellant Gimballed 

tanks manifold valves OMS engine 




FIGURE 6-7. Simplified sketch at the left aft pod of the Space Shuttle’s Orbiting Maneu- 
vering System (OMS) and the Reaction Control System (RCS). (Source: NASA.) 



period and use more propellant than the smaller tanks allow; it is also possible 
to feed propellant from the left aft system to the one on the vehicle’s right side, 
and vice versa. These features allow for more than nominal total impulse in a 
portion of the thrusters, in case it is needed for a particular mission mode or an 
emergency mode. 

The compartmented steel propellant tanks with antislosh and antivortex baf- 
fles, sumps, and a surface tension propellant retention device allow propellant 
to be delivered independent of the propellant load, the orientation, or the accel- 
eration environment (some of the time in zero-g). Gauges in each tank allow a 
determination of the amount of propellant remaining, and they also indicate a 
leak. Safety features include sniff lines at each propellant valve actuator to sense 
leakage. Electrical heaters are provided at propellant valves, certain lines, and 
injectors to prevent fuel freezing or moisture forming into ice. 

Some pressure feed systems can be prefilled with storable propellant and pres- 
surizing agent at the factory and stored in readiness for operation. Compared to 
a solid propellant rocket unit, these storable prepackaged liquid propellant pres- 
surized feed systems offer advantages in long-term storability, random restarts, 
and resistance to transportation vibration or shock. 



6.4. GAS PRESSURE FEED SYSTEMS 215 



TABLE 6-5. Characteristics of the Orbital Maneuver System (OMS) and the Reaction 
Control System (RCS) of the Space Shuttle in one of the Aft Side Pods 



Item 


OMS 


Primary RCS 


Vernier RCS 


Thrust (per nozzle) (Ibf) 


6000 


870 


25 


Number of thrusters per pod 


1 


12 


2 


Thrust chamber cooling 


Regenerative and radiation 


Radiation cooling 


Chamber pressure, nominal 


125 


152 


110 


(psi) 








Specific impulse (vacuum 


313 


280“ 


265“ 


nominal) (sec) 








Nozzle area ratio 


55 


22-30" 


20-50“ 


Mixture ratio (oxide/fuel mass 


1.65 


1.6 


1.6 


flow) 








Bum time, minimum (sec) 


2 


0.08 


0.08 


Bum time, maximum (sec) 


160 


150 


125 


Bum time, cumulative (sec) 


54,000 


12,800 


125,000 


Number of starts, cumulative 


1000 


20,000 


330,000 


(sec) 








Oxidizer (N 2 O 4 ) weight in 


14,866 


1464 


tank (lb) 








Fuel (MMH) weight in tank 


9010 


923 


(lb) 








Number of oxidizer/fuel tanks 


1/1 




1/1 


Propellant tank volume, each 


90 




17.9 


tank (ft^) 








Ullage volume, nominal (full 


7.8 


1.2 


-1.5 


tank) (ft^) 








Tank pressure, nominal (psi) 


250 


280 


Helium storage tank pressure 


4700 


3600 


(psi) 








Number of helium tanks 


1 




2 


Volume of helium tanks (ft^) 


17 




1.76 



“Depends on specific vehicle location and scarfing of nozzle. 
Sources: NASA, Aerojet General Corporation. 



The thrust level of a rocket propulsion system with a pressurized gas feed 
system is determined by the magnitude of the propellant flows which, in turn, 
is determined by the gas pressure regulator setting. The propellant mixture ratio 
in this type of feed system is controlled by the hydraulic resistance of the liq- 
uid propellant lines, cooling jacket, and injector, and can usually be adjusted by 
means of variable restrictors or fixed orifices. Further discussion of the adjust- 
ing of thrust and mixture ratio can be found in Sections 11.5 and 11.6 and in 
Example 11-2. 




- 5 5 o 4 



- E 2 ^ 



-o o 

C C3 
N 

2 -S 

at 

-5 

"S 

s 

0 
d 

1 



E 

D 

V 5 

00 



too 

c 



0 ) d) 

> c 

D ^ 



d c^ 
.■d T3 
X) . .. 

U /—N 

°d 

■£ Z 

^ CD 

^ "S 

^ n 

S 

"S 



a; eu 3 

^ P 



l-H . . 

0 !Z 3 

tM (U 

!C ^ 

1 ^ 
V 5 O 

T 3 S 

,D > 

C .-d 
d ^ 
X Vh 
D O 

o ii 

U ti 

G, S 
D X 
X C/5 

^ “ 

® d 

c 

u lU 

bJ) X 
ra •'“' 

0 ^ 

X ^ 

c^ 

X w 

^ r- 

X £ 
X D 

£ ^ 
•P c/5 
00 ^ 
O 

00 ^ 

1 ^ 

^ rS 



g 



216 



(MMH); short dashed lines: high-pressure helium. (Source: NASA.) 




6.5. TANK PRESSURIZATION 217 



6.5. TANK PRESSURIZATION 

As was stated before the objective of feed systems is to move the propellants 
under pressure from the propellant tanks to the thrust chamber(s). The tank pres- 
surization system is that part of the feed system that provides a gas to expel 
propellants from the tanks. See Refs. 6-1, 6-8, 6-9, and 6-10. As was described 
in Section 6.3 there are two types: (1) in a pressurized gas feed system a rel- 
atively high pressure gas displaces the propellants from the tanks and (2) in a 
pumped feed system (described in the next section) the main energy for feeding 
the propellants comes from one or more pumps, but it requires a low gas pressure 
in the tanks to move the propellants to the pump inlet in order to avoid pump 
cavitation. 

There are several sources of pressurizing gas commonly used in tank pressur- 
ization systems. 

1 . High-pressure inert gas stored at ambient temperature is the most common. 
Typical gases are helium, nitrogen, or air. Table 6-3 shows a comparison 
of the regulated pressure system (see Fig. 1-3) and the blow-down system 
(see Fig. 6-6). It is discussed further in this section. As the gas expands, 
its temperature drops. 

2. Heated high-pressure inert gas (typically 200 to 800°F or 93 to 427°C) 
reduces the amount of required gas mass and thus the inert mass of the 
pressurizing system. Examples are gas heated by a heat exchanger with 
warm gas from a gas generator, with warm gas from turbine exhaust, or 
with electrical heaters inside the gas tank. Reference 6-12. 

3. Gas is created by a chemical reaction by using either liquid bipropellants 
or a monopropellant and alternatively by a solid propellant, all at mixture 
ratios that allow a low enough or “warm” gas reaction temperature, so 
uncooled hardware can be used. The term “warm gas” (say 400 to 1600°F 
or 204 to 871°C) is used here to distinguish it from the hot gas (4000 to 
6000°F, or 2204 to 3319°C) in the main combustion chamber. This chem- 
ically generated warm gas usually results in a lighter tank pressurization 
system than the heated inert gas system, particularly for applications with a 
high total impulse. This gas can come from two separate small gas genera- 
tors for tank pressurization; one produces fuel-rich gas for pressurizing the 
fuel tank and the other feeds oxidizer-rich pressurizing gas into the oxidizer 
tank. This scheme was first used in the United States on the Bomarc rocket 
engine around 1952 and on the Russian RD-253 around 1961. The other 
common warm-gas scheme is to bleed a small amount of warm gas from the 
engine’s gas generator or preburner. If this gas is fuel rich, then it can only 
be used for pressurizing the fuel tank, and it may need to be cooled. These 
chemical reaction gases typically are at 1200 to 1700°F or 649 to 921°C, 
which are within the gas temperatures allowable for most alloy turbines and 
steel tanks. Catalyzed decomposed hydrazine warm gaseous reaction prod- 
ucts have successfully pressurized liquid hydrazine tanks. With aluminum 



218 



LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



propellant tanks it is necessary to cool the warm gas further. Many alu- 
minum alloys melt around 1100°F (593°C). This cooling of gas has been 
accomplished by using a heat exchanger with one of the propellants. Solid 
propellant gas generators have been used on experimental liquid propellant 
rocket engines, but as far as the author can determine, none have been 
adapted for a production flight vehicle. One clever system developed in 
the former Soviet Union uses two rocket engines operating simultaneously 
on the same vehicle. The larger engine has a bleed of oxidixer-rich gases 
from an oxidizer-rich preburner for pressurizing the common oxidizer tank. 
The second engine feeds four smaller hinge-mounted vernier thrust cham- 
bers, which hare been used for attitude control and also for extra thrust; 
its fuel-rich gas generator has a bleed of gas for pressurizing the common 
fuel tank. See Ref. 6-12. 

4. Evaporate a flow of a cryogenic liquid propellant , usually liquid hydrogen, 
by applying heat usually taken from a thrust chamber cooling jacket and 
then use a part or all of this evaporated flow for tank pressurization. Orifices 
or pressure regulators may be needed to control the desired tank pressure. 
This scheme is used for the fuel tank of the Space Shuttle Main Engine and 
the tap-off stub for pressurizing gas can be seen in Fig. 6-12 in the turbine 
exhaust manifold of the fuel booster pump. The oxygen tank is pressurized 
by gasified liquid oxygen, which was tapped off the discharge side of the 
main oxygen pump and heated in a heat exchanger around the turbine of 
the main oxygen pump, as shown in Figure 6-12. See Ref. 6-12. 

5. Direct injection of a small stream of hypergolic fuel into the main oxidizer 
tank and a small flow of oxidizer into the fuel tank has been tried in several 
countries without success. It is really a form of chemical gas generation. 
The only successful flight application is reported to be with a Russian 
ICBM upper stage engine using hypergolic propellants. See Refs. 6-1 and 
6 - 12 . 

6. Self-pressurization of cryogenic propellants by evaporation is feasible but 
can be difficult to control (see Ref. 6-3). Experience is limited. 

In order to design or analyze a pressurizing system it is necessary to know 
or to assume certain data about the tank and the engine. This can include the 
basic engine parameters, such as the propellant flow, thrust, duration, pulse width, 
propellant tank volume, percent ullage of that volume, storage temperature range, 
propellant and pressurizing gas properties, propellant tank pressure, gas tank 
pressure, or amount of unavailable residual propellant. Eor many of these the 
nominal, maximum, and minimum values may be needed. 

Factors Influening the Required Mass of Pressurizing Gas 

A key task in the analysis and design of tank pressurization systems is determining 
the required mass of pressurizing gas. Many different factors influence this gas 
mass, and some of them can be quite complex as shown in Ref. 6-1. Therefore 



6.5. TANK PRESSURIZATION 219 



a basic analysis of the amount of pressurizing gas for a new propulsion system 
is usually not accurate or fully useful. The mass of gas can be estimated using 
simplifying assumptions, but it is more accurate if it is based on actual test 
results and/or data from similar proven pressurizing systems. These are some of 
the influencing factors: 



1 . Evaporation of propellant at the interface between the pressurizing gas 
and the liquid propellant is a key phenomenon. The evaporated propellant dilutes 
the gas and changes its properties and expansion behavior. It depends on the 
temperature difference between the gas and the liquid, sloshing, vapor pressure 
of the propellant, turbulence, and local velocities of the gas. Furthermore the 
him of propellant on those portions of the tank walls and baffles, which are 
above the liquid level, will also evaporate. “Warm” gases (e.g., a bleed from a 
gas generator) will cause heating of the top layer of the liquid propellant and 
increase propellant evaporation. With cryogenic propellants the gas is always 
warmer than the liquid. The gas is cooled and more liquid evaporates. 

2. The temperature of those walls of the propellant tanks, which are part of the 
exterior vehicle surface, can be affected by aerodynamic heating, which varies 
during the flight. This heat can increase the gas temperature as well as the liquid 
propellant temperature, and can augment the liquid evaporation. 

3. The solubility of the gas in the liquid is affected by temperature and pres- 
sure. For example, nitrogen gas is soluble in liquid oxygen. It requires four times 
as much nitrogen gas to pressurize liquid oxygen as an equivalent volume of 
water. This dilutes the oxygen and causes a small performance loss. Helium is 
not very soluble in oxygen. 

4. Condensation of certain species of the gas can dilute the propellant. Water 
vapor in warm gases is an example. Condensation can also occur on the exposed 
wetted inner walls of the propellant tanks and this requires more pressurizing 
gas. 

5. Changes in the gas temperature take place during the operation. The com- 
pressed gas undergoes an expansion that can cause a dramatic cooling of the gas. 
Temperatures as low as 160 to 200 K (—228 to — 100°F) have been recorded 
with helium. The cold gas absorbs heat from the propellants and the engine 
hardware. The particular expansion process will depend largely on the time of 
rocket operation. For large liquid propellant rocket engines, which run only for 
a few minutes, the expansion process will be close to adiabatic, which means 
little heat transfer from the hardware to the gas. For satellites, which stay in 
orbits for years and where the thrusters operate only occasionally for short oper- 
ating periods, heat will be transferred from the vehicle hardware to the gas 
and the process will be close to an isothermal expansion (no change in tempe- 
rature). 

6. Chemical reactions of some species of the pressurizing gas with the liquid 
propellant have occurred and some can generate heat. Inert gases, such as helium, 
have no chemical reactions with the propellants. 



220 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



7. Turbulence and irregular flow distribution of the entering gas will increase 
the heat transfer between the liquid and the gas. Depending on the temperature 
difference, it can cause additional heating or cooling of the top liquid layers. 

8. Vigorous sloshing can quickly change the gas temperature. In some of 
the experimental flights of the Bomarc missile the side accelerations induced 
sloshing, which caused a sudden cooling of the warm tank pressurizing gas and 
caused a sudden reduction of tank pressure and propellant flow. See Refs. 6-4 
and 6-5. 

9. In many rocket engines some of the pressurizing gas is used for purposes 
other than tank pressurization, such as actuation of valves or controls. The amount 
required has to he determined and added to the total needed gas mass. 



Simplified Analysis for the Mass of Pressurizing Gas 

This is one example for a pressurizing system using a compressed gas initially 
in a separate tank at ambient temperature. It is the first category of pressurizing 
systems discussed above and it is perhaps the most common type. It assumes an 
adiabatic expansion of the gas (implies no heat transfer to or from the system 
hardware), the gas behaves like an ideal gas (obeys the gas laws), and the con- 
servation of energy can be applied. Since the tank often has some insulation and 
since the operation of the rocket engines is short, the expansion process will be 
close to an adiabatic one. The expansion of the gas in the storage tank causes a 
significant drop in gas temperature and an increase in gas density. Furthermore 
it assumes that there is no evaporation of the liquid propellant (usually valid for 
a propellant with low vapor pressure), the gas is inert, does not dissolve in the 
liquid propellant, and there is no sloshing or vortexing. 

Let the initial condition in the gas tank be given by subscript 0 and the 
instantaneous conditions in the gas tank by subscript g and in the propellant tank 
by subscript p. The gas energy after and before propellant expulsion is 

mgCvTg+mpCvTp+ppVp=moCyTo (6-5) 

The work done by the gas in displacing the propellants is given by PpVp. 
Using Eqs. 3-3 to 3-6, the initial storage gas mass nto may be found: 



-|- CyPpVp j R “f ppVp — uiqCdTq 
m = iPgVo+ppVpk)/{RTo) 



( 6 - 6 ) 



Here c„ is the specific heat at constant volume, R is the gas constant per unit 
mass, and k is the specific heat ratio. This equation may be expressed as 



Pgm Pp^P j. ^ FpFp / k 
Po RTq RTq \l-Pg/po 



(6-7) 



6.6. TURBOPUMP FEED SYSTEMS AND ENGINE CYCLES 221 



The first term in this equation expresses the mass of gas required to empty 
a completely filled propellant tank if the gas temperature is maintained at the 
initial storage temperature 7 q. The second term expresses the availability of the 
storage gas as a function of the pressure ratio through which the gas expands. 

Example 6-2. Determine the approximate mass and volume of air needed to pressurize 
a tank of 250 kg of 90% hydrogen peroxide for an initial gas tank pressure po of 14 MPa 
and a required propellant tank pressure of 3.40 MPa. The density of this propellant is 
1388 kg/m^, the gas constant R is 289 J/kg-K, the ambient temperature is 292 K, and for 
air ^ = 1 .40. 



SOLUTION. Since the vapor pressure of 90% hydrogen peroxide is low, the amount of 
liquid propellant that would evaporate is very small and the assumption of no evapora- 
tion can apply. The propellant volume is 250/1388 = 0.180 m^. Allow 5% more gas to 
compensate for the ambient temperature changes of the liquid propellant and the stored 
gas, and for geometric variations of the hardware. From Eq. 6-7 the initial mass of air 
mo is 



PpVp k 

mo = — — 

RTa 1 - (Ps/pq) 

3.4 X 10® X 0.180 X 1.05 x 1.40 
“ 289 X 298 X [1 - (3.4/14)] 

= 13.8 kg of compressed air 

The volume of the gas tank is from Eq. 3-4: 

Vo = nioRTo/po = 13.8 x 289 x 298/(14 x 10®) = 0.0847 m^ 

The high-pressure gas tank volume is almost half of the propellant tank volume. 



6.6. TURBOPUMP FEED SYSTEMS AND ENGINE CYCLES 

The principal components of a rocket engine with one type of turbopump system 
are shown in the simplified diagram of Fig. 1 -4. Here the propellants are pres- 
surized by means of pumps, which in turn are driven by turbines. These turbines 
derive their power from the expansion of warm gases. 

Figures 10-1, 10-2, and 10-3 show examples of turbopumps and Chapter 10 
is devoted exclusively to the turbopump. It is a high precision accurately bal- 
anced piece of high shaft speed (rpm) rotating machinery. It has usually two or 
one centrifugal pump and a turbine. Its high-speed high-load bearings support the 
shaft(s) on which the pump(s) and turbine are mounted. It has shaft seals to pre- 
vent leakage of propellants and also to prevent the two propellants from mixing 
with each other inside the turbopump. Some turbopumps also have a gear trans- 
mission, which allows the pumps or turbine to rotate at a different, usually more 



222 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



efficient, shaft speed. Chapter 10 describes the design of turbopumps, the several 
arrangements of the key components, the design of its major components, and 
alternate configurations. While pressurized feed systems can be started relatively 
quickly, the starting of turbopump feed systems usually takes longer, because it 
takes some time for the rotating components (pumps, turbines) to accelerate to 
operating shaft speed. Starting is discussed in Section 11.4. 

Engines with turbopumps are preferred for booster and sustainer stages of 
space launch vehicles, long-range missiles, and in the past also for aircraft 
performance augmentation. They are usually lighter than other types for these 
high-thrust, long-duration applications. The inert hardware mass of the rocket 
engine (without tanks) is essentially independent of duration. Examples can be 
seen in Eigs. 6-1 and 6-9 and also in Refs. 6-1, and 6-2. Eor aircraft perfor- 
mance augmentation the rocket pump was driven directly by the jet engine, as 
in Ref. 6-13. Erom the turbopump feed system options depicted in Chapter 10, 
the designer can select the most suitable concept for a particular application. 

As stated elsewhere in this book turbopump feed systems are usually preferred 
when the engine has a relatively high total impulse, which means high thrust and 
long duration. Pressurized feed systems are best for rocket engines with relatively 
low total impulse, that is, low thrust and often low cumulative firing durations. 

Engine Cycles 

Engine cycles apply only to liquid propellant rocket engines with a turbopump 
feed system. The three most common engine cycles are shown in Eig. 6-9 and 
many of each of these have flown. Reference 6-14 shows variations of these 
three cycles and other feasible cycles; some of them have not yet flown and 
some have not yet even been built. 

An engine cycle for turbopump-fed engines describes the specific propellant 
flow paths through the major engine components, the method of providing the hot 
gas to one or more turbines, and the method of handling the turbine exhaust gases. 
There are open cycles and closed cycles. Open denotes that the working fluid 
exhausting from the turbine is discharged overboard, usually after having been 
expanded in a separate nozzle of its own as in Eigs. 1-4 and 6-10, or discharged 
into the nozzle exit section of the thrust chamber at a point in the expanding 
section far downstream of the nozzle throat as shown in Eig. 6-9. In closed 
cycles or topping cycles all the working fluid from the turbine is injected into 
the engine combustion chamber to make the most efficient use of its remaining 
energy. In closed cycles the turbine exhaust gas is fed into the injector of the 
thrust chamber and is expanded through the full pressure ratio of the main thrust 
chamber nozzle, thus giving a little more performance than the open cycles, 
where these exhaust gases expand only through a relatively small pressure ratio. 

Table 6-6 shows key parameters for each of the three common cycles and 
it describes the differences between them. The schematic diagrams of Eig. 6-9 
show each cycle with a separate turbopump for fuel and for oxidier. However, 
an arrangement with the fuel and oxdizer pump driven by the same turbine is 



6.6. TURBOPUMP FEED SYSTEMS AND ENGINE CYCLES 223 






FIGURE 6-9. Simplified diagrams of three engine cycles for liquid propellant rocket 
engines. The spirals are a symbol for an axisymmetric cooling jacket where heat is 
absorbed. 



also feasible and sometimes reduces the hardware mass, volume, and cost. The 
“best” cycle has to be selected on the basis of the mission, the suitability of 
existing engines, and the criteria established for the particular vehicle. There is 
an optimum chamber pressure and an optimum mixture ratio for each application, 
engine cycle, or optimization criterion, such as maximum range, lowest cost, or 
highest payload. 

The gas generator cycle has been the most common. Compared to other engine 
cycles, it is relatively simple, pressures are usually lower, generally has a lower 
inert mass, and its engine cost is usually lower. However, its performance (spe- 
cihc impulse) is lower by a few percent. Its performance is adequate for many 
space flight and military missions. 



TABLE 6-6. Qualitative Characteristics for Three Different Engine Cycles 



u 

c 

o 



D 

S 

o 

U 

00 

c 3 



00 

o 

I 

CN 

o 



o 



o 



o 

o 

I 

o 



^ I 

^ o 
*3! 



.S 

£ 

o 

.£ 

Hi 

pH 



3 o 

X 

£ 3 



(D ^ 

5 

00 



Oh (D 
? 

00 



OJ Oh . 

a 

3 o 

=2 '3 

00 g 

Gi 



M3 

o^ 



in D 
ON Cd 

-H (D 



CQ 

c« ^ 

^ e 

c§ 

^ &n 
„ - O 

a Q 
z > 
a -3 
o 



o 

a 



0^ 

U 



•o 

c 

Oh 

X 

a 



M3 

O 

I 

CM 

O 



o ^ 

13 o 

o 



<D 

x: 



o o 
cn 



HH ^ ' 

o 

0 

^ (N 

M3 ^ 

ON 

1 

in 

i> 



00 

c 



o 

o 

CN 

I 

o 

»n 



00 

o 



.£ 

3 

£ 

o 

.£ 

~o 

o 

PP 



-a 

o 

3 

f/3 

cO o 

00 c 

(D X 

1 3 

J-H f/3 . i-H 

O ^ T3 O 

00 ® o Of 

X3 f/3 O ‘X 

® S 13 ® 

& S 

O 

O 

S 3 
U 



O CO 
NO NO 
ON ON 



O O 

c c 
3 3 

o o 
-1^ 

CJ CJ 

o o 
D< Di 

>, >> 

0 o 

3 3 
3 3 

1 ^ 



&H CH 



>. 

u 



o 

c/3 

cO 

a 



o 

o 



l> 



o 

ON 

I 

o 



^ o 

^ I 

^ in 

a rn 



■S) s 

o a 

o C!h 
CO 
Vh 

t3 O 
O 

g 

-P N 



O 

Q 



cO 

O. 

<D 



I !§ 

^ O 
(D cO 

Z 3 

3 (U 

u 0< 
Pi 



I I 

Sh O 
QJ GS 

:>. 

O P-* 
00 o 

C« > 
00 

r- ^ 

•S Pi 
o 



CO 



m E 

ON o 
^ U ON 

<, (D ON 
C/5 ^ ^ 

O 3 ^ 

"O £ 

53 -S I 

i§ I 

.9 O 

O 

E 



U 

o 

c 

*ob 

c 

a 






3 O 

3 3 ?5§ 

£ 3 ^ 

0^ c/3 
CJ CJ CO 

CO • ^ 

■n C > 

CJ 01 

c/3 c/3 

... cO o 

c “.S 
M 'o ■£ 

C 3 

W H 



u 

XI 

E 



■3 X 
3 ° 



Oh CO 
O CJ ’ 

Oh Oh 



o 



Oh O 00 



(U 



CJ 



c 

M 3 ”0 

c/3 >— I 
O O Oh 
l-i Oh O 
Oh o § 

S-H O 

Oh Oh 



£ o 

I s 

i.s 

-C Xi 



o 

c 

'oo 

c 

o 



T3 

O 



Oh 



c 

.2 

“ to 

c/3 }_ 

(U O 
Oh 

•o o 

C 

3 x: 
O 00 

t-( 

00 <n 

‘ tfl tfl 
Vh U, 

3 3 



224 



Rossiyskiy Naucho Isseldovatelskiy Institut (Reaction Propulsion Research Institute). 



6.6. TURBOPUMP FEED SYSTEMS AND ENGINE CYCLES 225 



In the gas generator cycle the turbine inlet gas comes from a separate gas 
generator. Its propellants can be supplied from two separate propellant tanks or 
can be bled off the main propellant feed system tanks. Some early engines used a 
separate monopropellant for creating the generator gas. The German V-2 missile 
engine used hydrogen peroxide, which was decomposed by a catalyst. Typically, 
the turbine exhaust gas is discharged overboard through one or two separate small 
low-area-ratio nozzles (at relatively low specific impulse), as shown schematically 
in Fig. 1-4 and in the Vulcain engine or RS-68 engine listed in Table 11-2. See 
Ref. 6-15. Alternatively, this turbine exhaust can be aspirated into the main flow 
through multiple openings in the diverging nozzle section, as shown schematically 
in Fig. 6-9 for a gas generator engine cycle. This gas then protects the walls near 
the nozzle exit from high temperatures. Both methods can provide a small amount 
of additional thrust. The gas generator mixture ratio is usually fuel rich (in some 
engines it is oxidizer rich) so that the gas temperatures are low enough (typically 
900 to 1350 K) to allow the use of uncooled turbine blades and uncooled nozzle 
exit segments. The RS-68 rocket engine, shown in Fig. 6-10, has a simple 
gas generator cycle. As can be seen from the data under the figure, with a gas 
generator cycle the specific impulse of the thrust chamber by itself is always a 
little higher than that of the engine, and the thrust of the thrust chamber is always 
slightly lower than that of the engine. The thrust of the RS-68 can be reduced or 
throttled to less then 60% of full thrust. See Ref. 6-15. 

The expander cycle and the staged combustion cycle are both closed cycles, 
and they offer several percent higher performance than the gas generator cycle; 
this small improvement makes a substantial difference in payload for flight mis- 
sions with high mission velocities. Alternatively they allow a somewhat smaller 
flight vehicle. However, the engines are usually more complex, heavier, and more 
expensive. 

A flow diagram of an expander cycle is shown in Fig. 6-11. The engine 
coolant (usually hydrogen fuel) is evaporated, heated, and then fed to low- 
pressure-ratio turbines after having passed through the cooling jacket where it 
picked up energy. Part of the coolant, perhaps 5 to 15%, bypasses the turbine 
(shown in Fig. 6-11) and rejoins the turbine exhaust flow before the entire 
coolant flow is injected into the engine combustion chamber where it mixes and 
bums with the oxidizer (see Refs. 6-14 and 6-16). The primary advantages of 
the expander cycle are good specific impulse, relative engine simplicity, no gas 
generator, and relatively low engine mass. In the expander cycle all the propel- 
lants are fully burned in the engine combustion chamber and expanded efficiently 
in the exhaust nozzle of the thrust chamber. 

This cycle is used in the RLIO hydrogen/oxygen rocket engine, and different 
versions of this engine have flown successfully in the upper stages of several 
space launch vehicles. Data on the RLIO engines are given in Tables 6-8, 8-1, 
and 11-2. A modification of this engine, the RLlOB-2 with an extendible nozzle 
skirt, can be seen in Fig. 8-17. The RLlOB-2 flow diagram in Fig. 6-11 shows 



226 LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 




Roll control 
nozzle using 
turbine 
exhaust 



Gimbal actuator 



Oxidizer valve 



Regenerative cooled thrust 
chamber with nozzle throat 

Ablative lined 
bell-shaped nozzle 
extension 



High pressure 
oxygen line 



Oxygen 

turbopump 



Oxygen tank 

pressurization 

heat 

exchanger 



Turbine 

exhaust 

nozzle 



Main fuel 
valve 



Eour leg structural 
support 



Fuel turbopump 



Parameter 


Thrust chamber 


Engine 


Specific impulse at sea level (max.), sec 


363 


357 


Specific impulse in vacuum (max.), sec 


415 


409 


Thrust, at sea level, Ibf 


640,700 


656,000 


Thrust in vacuum Ibf 


732,400 


751,000 


Mixture ratio 


6.74 


6.0 



FIGURE 6-10. Simplified view of the RS-68 rocket engine with a gas generator cycle. 
For engine data see Table 10-3. (Courtesy of Pratt & Whitney, Rocketdyne.) 



its expander cycle. The turbine drives a single-stage liquid oxygen pump (through 
a gear case) and a two-stage liquid hydrogen pump. The cooling down of the 
hardware to cryogenic temperatures is accomplished by flowing (prior to engine 
start) cold propellant through cool-down valves. The pipes for discharging the 
cooling propellants overboard are not shown here but can be seen in Fig. 8-17. 



6.6. TURBOPUMP FEED SYSTEMS AND ENGINE CYCLES 227 




Oxidizer flow 
control valve 



Oxygen pump 



Liquid 

oxygen 



Propellant 
shutoff valves 



Liquid 

hydrogen 



Fuel pump 



/ 



Fuel cooldown and 
pressure relief valve 



FIGURE 6-11. Simplified schematic flow diagram of the RLlOB-2 upper-stage rocket 
engine is an example of an expander engine cycle. For data see Tables 6-8 and 8-1. 
(Courtesy of Pratt & Whitney Rocketdyne.) 



Thrust is regulated by controlling the flow of hydrogen gas to the turbine, using 
a bypass to maintain constant chamber pressure. Helium is used as a means of 
power boost by actuating several of the larger valves through solenoid-operated 
pilot valves. 

In the staged combustion cycle, the coolant flow path through the cooling 
jacket is the same as that of the expander cycle. Here a high-pressure preburner 
(a high-pressure gas generator) burns all the fuel with part of the oxidizer to 
provide high-energy gas to the turbines. The total turbine exhaust gas flow is 
injected into the main combustion chamber where it burns with the remaining 
oxidizer. This cycle lends itself to high-chamber-pressure operation, which allows 
a small thrust chamber size. The extra pressure drop in the preburner and turbines 
causes the pump discharge pressures of both the fuel and the oxidizer to be higher 
than with open cycles, requiring heavier and more complex pumps, turbines, and 
piping. The turbine flow is relatively high and the turbine pressure drop is low, 
when compared to other cycles. The staged combustion cycle gives high specific 
impulse, but it is more complex and heavy. A variation of the staged combustion 
cycle is used in the Space Shuttle Main Engine, as shown in Figs. 6-1 and 6-12. 
This engine actually uses two separate preburner chambers, each mounted directly 
on a separate main turbopump. In addition, there are two more turbopumps for 
providing a boost pressure to the main pumps, but their turbines are not driven by 
combustion gases; instead, high-pressure liquid oxygen drives one booster pump 
and evaporated hydrogen drives the other. The injector of this reusable liquid 
propellant high-pressure engine is shown in Fig. 9-6 and performance data are 



228 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 




Low pressure 

Hydrogen t fuel (booster) ^ Low pressure t Oxygen 

^riy^l S ? iiv\/nQn riQo ' fc- ~ . ■ ■ 



Hydrogen gas bleed_^ 
for pressurizing | ' 
the fuel tank ^ 



One of two 
oxygen control 
valves of the 
preburners 



Thrust chamber 
gas exhaust 



FIGURE 6-12. Flow diagram illustrating the staged combustion cycle of the Space 
Shuttle Main Engine (SSME) using liquid oxygen and liquid hydrogen fuel. (Courtesy of 
Pratt & Whitney Rocketdyne and NASA.) 



given in Tables 10-1 and 11-2. While the Space Shuttle Main Engine (burning 
hydrogen with oxygen) has fuel-rich preburners, oxidizer-rich preburners are used 
in the Energomash RD-170 and RD-120 engines (kerosene/oxygen) and other 
Russian staged combustion rocket engines. See Table 11-3. Another example 
of a staged combustion cycle is the Russian engine RD-253; all of the nitrogen 
tetroxide oxidizer and some of the unsymmetrical dimethyl hydrazine fuel are 
burned in the preburner, and the remaining fuel is injeeted directly into the main 
combustion chamber. 

In an engine with a gas generator cycle the thrust of the thrust chamber is 
always slightly less than the Thrust of the engine, because the exhaust from the 
turbine provides a little extra thrust. Also the thrust chamber specihe impulse 
is a little higher than the rocket engine specific impulse, because the low-flow 
gas exhausted from the turbine has a very low specific impulse. Eor closed 
cycles, namely the expander cycle and the staged combustion cycle, the values 
of the thrust and the specific impulse are the same for the engine and its thrust 
chamber(s). 



6.7. ROCKET ENGINES EOR MANEUVERING, ORBIT ADJUSTMENTS, OR ATTITUDE CONTROL 229 



6.7. ROCKET ENGINES FOR MANEUVERING, ORBIT ADJUSTMENTS, 
OR ATTITUDE CONTROL 

These engines have usually a set of small thrusters that are installed at various 
places in a vehicle and a common pressurized feed system, similar to Figs. 1-3, 
4-13, or 6-13. They are called reaction control systems or auxiliary rocket 
propulsion systems as contrasted to higher-thrust primary or boost propulsion 
systems. Most have low thrust, use storable liquid propellants, require accurate 
repeatable pulsing, a long life in space, and/or a long-term storage with loaded 
propellants in flight tanks. Typical thrust levels of a small thruster are between 0. 1 
and 1000 Ibf. Figure 4-13 shows that it requires 12 thrusters for the application of 
pure torques about three vehicle axes. If a three-degree-of-rotation freedom is not 
a requirement, or if torques can be combined with some translation maneuvers, 
fewer thrusters will be needed. These auxiliary rocket engines are commonly 
used in spacecraft or missiles for the accurate control of flight trajectories, orbit 
adjustments , or attitude control of the vehicle. References 6-1 and 6-17 give 
information on several of these. Figure 6-13 shows a simplifled flow diagram 
for a postboost control rocket engine, with one larger rocket thrust chamber for 
changing the velocity vector and eight small thrusters for attitude control. 

Section 4.5 describes various space trajectory correction maneuvers and satel- 
lite station-keeping maneuvers that are typically performed by these small aux- 
iliary liquid propellant rocket engines with multiple thrusters. Table 6-7 lists 
typical applications for rocket engines with small thrusters. 

Attitude control can be provided both while a primary propulsion system (of 
a vehicle or of a stage) is operating and while its small thruster rocket system 
operates by itself. For instance, this is done to point satellite’s telescope into 



TABLE 6-7. Typical Applications for Small Thrusters 

Flight path (or orbit) corrections or changes 
Minor flight velocity adjustments 
Station keeping (correcting for deviations from orbit) 

Orbit injection for small satellites 
Deorbit maneuver for small satellites 
Divert maneuvers of terminal interceptors 
Attitude control for: 

Satellites, stages of space launch vehicles, space stations, missiles 
Roll control for a single gimbaled rocket engine 
Pointing/orienting antennas, solar cells, mirrors, telescopes, etc. 

Correct the misalignment of principal, larger thrust chamber 
Velocity tuning of war heads (postboost control system) for accurate targeting 
Settling of liquid propellants prior to gravity-free start of main upper stage engine 
Flywheel desaturation 



Source: Mostly from Ref. 6-17. 



230 LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 



a specific orientation or to rotate a spacecraft’s main thrust chamber into the 
desired direction for a vehicle turning maneuver. 

A common method for achieving accurate velocity corrections or precise angu- 
lar positions is to operate or fire some of the thrusters in a pulsing mode (e.g., 
fire repeatedly for 0.020 sec, each time followed by a pause of perhaps 0.020 to 
0.100 sec). The guidance system determines the maneuver to be undertaken and 
the vehicle control system sends command signals to specific thrusters for the 
number of pulses needed to accomplish this maneuver. Small liquid propellant 
engine systems are uniquely capable of these pulsing operations. Some small 
thrusters have been tested for more than 300,000 pulses. For very short pulse 
durations the specific impulse is degraded by 5 to 25%, because the performance 
during the thrust buildup and thrust decay period (at lower chamber pressure) is 
inferior to operating only at the rated chamber pressure and the transient time 
becomes a major portion of the total pulse time. 

Ballistic missile defense vehicles usually have highly maneuverable upper 
stages. These require substantial side forces also called divert forces (200 to 
6000 N) during the final closing maneuvers just prior to reaching the target. 
In concept the system is similar to that of Fig. 6-13, except that the larger 
thrust chamber would be at right angles to the vehicle axis. A similar system for 
terminal maneuvers, but using solid propellants, is shown in Figs. 12-27 and 
12-28. 

The Space Shuttle performs its reaction control with 38 different thrusters, 
as shown schematically in Figs. 1-14 and 6-8; this includes several duplicate 
(spare or redundant) thrusters. Selected thrusters are used for different maneuvers, 
such as space orbit corrections, station keeping, or positioning the Space Shuttle 
for reentry or visual observations. These small restartable rocket engines are also 
used for space rendezvous or docking maneuvers , where one spacecraft slowly 
approaches another and locks itself to the other, without causing excessive impact 
forces during this docking manuever. This docking operation requires rotational 
and translational maneuvers from a series of rocket engines. 

The application of pure torque to spacecraft can be divided into two classes, 
mass expulsion types (rockets) and nonmass expulsion types. Nonmass expulsion 
types include momentum storage, gravity gradient, solar radiation, and magnetic 
systems. Some space satellites are equipped with both the mass and nonmass 
expulsion types. Reaction wheels or flywheels, a momentum storage device, are 
particularly well suited to obtaining vehicle angular position control with high 
accuracies of less than 0.01° deviation and low vehicle angular rates of less 
than 10“^ degrees/sec with relatively little expenditure of energy. The vehicle 
angular momentum is changed by accelerating (or decelerating) the wheel. Of 
course, when the wheel speed reaches the maximum (or minimum) permissible, 
no further electrical motor torquing is possible; the wheel must be decelerated (or 
accelerated) to have its momentum removed (or augmented), a function usually 
accomplished through the simultaneous use of two small attitude control rocket 
thrusters, which apply a torque to the vehicle in the opposite direction. This has 
been called desaturation of the fly wheel. 



6.7. ROCKET ENGINES EOR MANEUVERING, ORBIT ADJUSTMENTS, OR ATTITUDE CONTROL 231 




FIGURE 6-13. Schematic flow diagram of the helium-pressurized, bipropellant rocket 
engine system of the fourth stage of the Peacekeeper ballistic missile, which provides 
the terminal velocity (in direction and magnitude) to each of several warheads. It has 
one larger gimballed thrust chamber for trajectory translation maneuvers and eight small 
thrusters (with scarfed nozzles) for attitude control in pitch, yaw, and roll. For clarity the 
tanks and feed system are shown outside the vehicle skin; really they are located within 
the vehicle. (Courtesy of USAF.) 



232 LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 



The propellants for auxiliary rockets fall into three categories: cold gas jets 
(also called inert gas jets), warm or heated gas jets, and chemical combustion 
rockets, such as bipropellant liquid propellant rockets. The specific impulse is 
typically 50 to 120 sec for various cold gas systems and 105 to 250 sec for 
warm gas systems. Warm gas systems can use inert gas with an electric heater 
or a monopropellant which is catalytically and/or thermally decomposed. Bipro- 
pellant attitude control thrust chambers allow an /j of 220 to 325 sec and have 
varied from 5 to 4000 N thrust; the highest thrusts apply to large spacecraft. All 
basically use pressurized feed systems with multiple thrusters or thrust cham- 
bers equipped with fast-acting, positive-closing precision valves. Many systems 
use small, uncooled, metal-constructed thrusters with supersonic exhaust nozzles, 
and they are strategically located on the periphery of the spacecraft pointing in 
different directions. Gas jets are used typically for low thrust (up to 10 N) and 
low total impulse (up to 4000 N-sec). They have been used on smaller satellites 
and often only for roll control. See Ref. 6-17 and Section 7.6. 

Small liquid monopropellant and liquid bipropellant rocket units are common 
in auxiliary rocket systems for thrust levels typically above 2 N and total impulse 
values above 3000 N-sec. Hydrazine is the most common monopropellant used 
in auxiliary control rockets; nitrogen tetroxide and monomethylhydrazine is a 
common bipropellant combination. The next chapter contains data on all three 
categories of these propellants, and Chapter 8 shows diagrams of small thrusters. 

Combination systems are also in use. Here a bipropellant with a relatively high 
value of Is, such as N2O4 and N2H4, is used in the larger thrusters, which consume 
most of the propellant; then several simple monopropellant thrusters (with a 
lower Is), used for attitude control pulsing, usually consume a relatively small 
fraction of the total fuel. Another combination system is to employ bipropellant 
or monopropellant thrusters for adding a velocity increment to a flight vehicle 
or to bleed or pulse some of the pressurizing gas, such as helium, through small 
nozzles controlled by electromagnetic valves to provide roll control. The specific 
mission requirements need to be analyzed to determine which type or combination 
is most advantageous for a particular application. 

Special thruster designs exist that can be used in a bipropellant mode at higher 
thrust and also in a monopropellant mode for lower thrust. This can offer an 
advantage in some spacecraft applications. An example is the Northrop-Grumman 
secondary combustion augmented thruster (SCAT), which uses hydrazine and 
nitrogen tetroxide, is restartable, vaporizes the propellants prior to injection and 
therefore has very efficient combustion (over 99%), can operate over a wide range 
of mixture ratios, and can be throttled from 5 to 15 Ibf thrust. From Ref. 6-12. 



6.8. ENGINE FAMILIES 233 



6.8. ENGINE FAMILIES 

An engine family is a series of related rocket engines, which have evolved over a 
period of many years. They come from the same rocket engine organization and 
each engine has been tailored to a specific application. The engines in the family 
have a strong resemblance to each other, use the same engine concept, usually 
the same propellants, and some identical or somewhat modified components of 
the same type. When an existing proven liquid propellant rocket engine can be 
modified and/or up-rated to fit a new application, the newer modified engine can 
use a lot of the proven hardware, test data, and software of the original earlier 
engines. 

An example is the RL 10 rocket engine family of upper-stage engines. It was 
developed by Pratt & Whitney Rocketdyne over a period of about 46 years and 
is shown in Table 6-8. Data is from Refs. 6-12 and 6-16. Each engine is for 
a specific application and is a modification and/or uprating of an earlier model. 
The principal changes from one engine model to the next included increases in 
thrust, increases in performance (somewhat higher specific impulse) by using 
higher chamber pressure, improved injector designs, and increases in nozzle exit 
area ratios. All use LOX/LH 2 propellants, the same basic engine concept with an 
expander engine cycle, the same tubular cooling jacket approach for the chamber 
and nozzle throat region, the same generic geared turbopump arrangement, often 
the same or similar valves, and power level control by a bypass of hydrogen gas 
around the turbine. In the turbopump the fuel pump and turbine are on the same 
high-speed shaft and the LOX pump is driven efficiently through a gear train at 
a lower speed. All engines are gimbal mounted (most at 4° maximum deflection) 
and most have space restart capability. Figure 8-17 shows the extendable nozzle 
of the RL lOB-2. In the RL 10A-3-3A thrust chamber a high-conductivity silver 
ring was brazed with silver into the nozzle throat, thus enabling a higher chamber 
pressure. The cooling jacket tubes were brazed with silver and were compatible. 
Figure 6-11 shows a flow sheet of an RL 10 expander engine cycle. 

It is interesting to see how thrust or specific impulse changed with time and 
with the model. Incidentally the specific impulse listed in Table 6-8 for the 
RL lOB-2 is the highest of any flying liquid propellant rocket engine in the world 
and the extendable nozzle exit segment of the RL lOA-4 was the first for liquid 
propellant rocket engines. 

When compared to a brand new engine, the principal benefits of adopting 
a modified engine, which is based on an earlier family of proven engines, are 
savings in costs (costs of design, development, less fabrication, less testing, quali- 
fication, and operation), attaining a high engine reliability more quickly, and often 
also a shorter schedule. The heritage of earlier proven similar engines allows 
the use of old engine or component data, having trained experienced personnel, 
proven subcontractors, a higher confidenee level of reliability, and often, but not 



Engine Qualification pi Thrust Mixture Weight 4 (vac) Nozzle No. Engines 

Model Year Vehicle psia Ibf Ratio Ibf sec Area Ratio per Stage Comments 



o. 

C £ 

0 W 

H s 

> -c . 

cj m 

o § 3 

^ — ,«5 
0) ^ 

CQ C/!) 

a « o 
^ > o 
C i- 

^ I “ 

1 s I 

IC (J x: 

0) 

Z 



D. 

£ 



o 

c 

_£ c 
o % 
^ o 

D 7D 

"o 
5S O 
P. ^ 
1) 

« g 

C3 



ffl 



U j, 

Q, Sj 

Q. c/n 



C 

— O 
"O 
a O 
a. o 



c 

0) 

u ■ 

>. • 



I ~a 
' -p 

■s 



Cl *- 

s e. 

3 u. 
Cl s 
O 

"P e 



O 

E- 



.2 N 



o OJ • — 



V. 



<D 

X 



00 



g- 00 

B 



<D 

X3 

c ^ 52 
£ > -g 
3 rp 
Q. C/5 C 



U S 
5 D 



t: 
c 

u -a S 



D H 
■d 



?3 lT 

La — /-s 



C W) 
1> a 
c >» 

'§ :> ' 

S "O ' 



1) "O 



^ C 



o ^ 
.5 X) 



-5 Z 



o 

> .£ 
C, 

00 > 



> o 

^ ® 3 ^ 

•a -c 2 
S o 

§ ° s- 

■ 3 II 8 
' s S 6 f 
S .S 2 S 



C 

U 



o ^ 
^ (50 

■K s= 

■£ -c 

(• i 



o 

o 



q 

X) 



(N 






(N 

00 

(N 





La 




La 


3 

3 

C 




La 

3 


3 

3 


3 

3 


> 


3 

3 


0) 

U 


C 

3 


3 

G 


C 


C 




C 


c 




(U 


O 

U 


0) 

U 


c 


0) 

U 


3 


H 


u 


c/5 


GO 


3 


GO 






— 


— 


— 


3 

00 


— 




c/5 

_3 


3 


< 


< 




< 






X 












< 


C/5 



u 



< 

o 



< 

o 



< 

o 






< 

o 



< 

o 



< 

o 



234 



RL 10A-3-3B 1986 Shuttle Centaur 425 15,000 6.0;1 310 436 61.0;1 2 RL 10A-3-3A modified to handle long 

space stay time for AF version of 
Shuttle/Centaur. Mixture ratio 
increased to 6.0 and cooldown 
components changed. Never flew. 



3 

H 



c 

.2 

c/3 

>» c 



0 ) O "O 



-D 

^ B 

TD X 
O 

a 

a. 

S 

9- S 






0) 

Z 



C3 ^ 

Ui 

O —H 
£■ ^ 
® J - 

o < 



•o > 

U - r - 

> cs 



-S OJ 5= 

w ^ .2 

Q. ^ c« 

^ e C 
^ 

^ ^ X 
tn 0 ) 



o .O- 
•o U 
cn *2 
^ 1j 
O Q 

B 

rr. W 



z 

c 

^ c 
t ? ^ 

3 O 
J= 'C 



^ .3 

■s a 



00 C/3 c/3 

3 G C 

■ 5--S s 

S g 

55 cc o 

0 « ;S "O 



I 

3 

J= 

u 



. 2P S 

-C o 
•o 



o 

•o 



3 o c 
c ^ 
§ cn g 



N 3 
O « 

c ® 

0) (U 

1 1 
3 TD 
-C C 



>< C > 



(N 

O ^ 



<N 

b> 

O 



p 



p 



lo 

d ^ 

lO 



m IT ) 
(N O" 
(N (N 



O 

'O 



3 3 

C G 



u 


u u 


u 


on 


c/3 C 


c/5 


— 




— 


< 


< P 


< 




■3" 0^ 


o 



U 

Q 



Q 



Qd 



< 

o 



< < 
o o 



<N 

< 

o 



< 

o 



CQ 

o 



235 



236 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



always, the use of the same debugged materials, fabrication, and test facilities. 
An intangible benefit is that the vehicle developer or prime contractor, who plans 
to use one of these engines, will have more confidence in the engine. A brand 
new engine may have slightly better performance and/or a somewhat lower inert 
engine mass and some other improvements, but it will usually be more costly, 
take longer to develop, and take more time to reach an equivalent high level of 
reliability. 



6.9. VALVES AND PIPELINES 

Valves control the flows of liquids and gases and pipes conduct these fluids to 
the intended components. There are no rocket engines without them. There are 
many different types of valves. All have to be reliable, lightweight, leakproof, 
and must withstand intensive vibrations and very loud noises. Table 6-9 gives 
several key classification categories for rocket engine valves. Any one engine 
will use only some of the valves listed here. 

The art of designing and making valves is based, to a large extent, on expe- 
rience. A single chapter cannot do justice to it by describing valve design and 
operation. References 6-1 and 6-2 decribe the design of specific valves, lines, 
and joints. Often the design details, such as clearance, seat materials, or opening 
time delay present development difficulties. With many of these valves, any sig- 
nificant leakage or valve failure can cause a failure of the engine itself. All valves 
are tested for two qualities prior to installation; they are tested for leaks — through 
the seat and also through the glands — and for functional soundness or perfor- 
mance. 

The propellant valves in high-thrust units handle relatively large flows at high 
service pressures. Therefore, the forces necessary to actuate the valves are large. 
Hydraulic or pneumatic pressure, controlled by pilot valves, operates the larger 
valves; these pilot valves are in turn actuated by a solenoid or a mechanical 
linkage. Essentially this is a means of power boost. 

Two valves commonly used in pressurized feed systems are isolation valves 
(when shut, they isolate or shut off a portion of the propulsion system) and latch 
valves; they require power for brief periods during movements, such as to open 
or shut, but need no power when latched or fastened into either an open or a 
closed position. 

A very simple and very light valve is a burst diaphragm . It is essentially a 
circular disk of material that blocks a pipeline and is designed so that it will 
fail and burst at a predetermined pressure differential. Burst diaphragms are 
positive seals and prevent leakage, but they can be used only once. The German 
Wassetfall anti-aircraft missile, an early application, used four burst disks; two 
were in high-pressure air lines and two were in the propellant lines. 



6.9. VALVES AND PIPELINES 237 



TABLE 6-9. Classification of Valves Used in Liquid Propellant Rocket Engines" 

1 . Fluid : fuel; oxidizer; cold or heated pressurized gas; hot turbine gas 

2. Application or Use: main propellant control; thrust chamber valve (dual or single); 
bleed; vent; drain; fill; by-pass; preliminary stage flow; pilot valve; safety valve; over- 
board dump; regulator; gas generator or preburner control; sequence control; prevent 
back flow; isolation of propellant or, latch valve 

3. Mode of Actuation: automatically operated (by solenoid, pilot valve, trip mechanism, 
pyrotechnic, etc.); manually operated; pressure-operated by air, gas, propellant, or 
hydraulic fluid (e.g., check valve, tank vent valve, pressure regulator, relief valve), with 
or without position feedback, rotary or linear actuator 

4. The Flow magnitude determines the size of the valve 

5. Duty cycle: single operation or multiple operation during the same flight, short duration 
pulse operation; reusable for other flights; long or short life 

6. Valve Type: normally open; normally closed; normally partly open; two-way; three- 
way, with/without valve position feedback; ball valve, gate valve, butterfly type, spring 
loaded, low pressure drops 

7. Temperature and pressure allow classification by high, low, or cryogenic temperature 
fluids, or high or low pressure or vacuum capability 

8. Accessible or not accessible to inspection, servicing, or replacement of valve or its seal 

"This list is not comprehensive or complete. 



Figure 6-14 shows a large main liquid oxygen valve. It is normally closed, 
rotary actuated, cryogenic, high pressure, high flow, reusable ball valve with a 
low-pressure loss in the open position, allowing continuous throtting, a controlled 
rate of opening through a crank and hydraulic piston (not shown), with a position 
feedback and anti-icing controls. 

Pressure regulators are special valves that are used frequently to regulate 
gas pressures. Usually the discharge pressure is regulated to a predetermined 
standard pressure value by continuously throttling the flow, using a piston, flexible 
diaphragm, or electromagnet as the actuating mechanism. Regulators can be seen 
in the flow sheets of Figs. 1-3 and 6-13. 

The various fluids in a rocket engine are usually transported by pipes or 
lines, usually made of metal and joined by fittings or welds. Their design must 
provide for thermal expansion and provide support to minimize vibration effects. 
For gimballed thrust chambers it is necessary to provide flexibility in the piping 
to allow the thrust axis to be rotated through a small angle, typically ±3 to 
±10°. This flexibility is provided by flexible pipe joints and/or by allowing 
pipes to deflect slightly when using two or more right-angle turns in the lines. 
The high-pressure propellant feed lines of the SSME have both flexible joints 
and right-angle bends, as shown in Figs. 6-1 and 6-15. This joint has flexible 
bellows as a seal and a universal joint-type mechanical linkage with two sets of 
bearings for carrying the separating loads imposed by the high pressure. 



238 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 




Valve 

outlet 



Shaft seal 

bearing 

Intergral 

ball/shatt/cams 



Bearing 



Inlet 

seal 



Cam follower 
pin and bearing 



Section A-A 



Hydraulic 

actuator 

housing 



Thermal. 

Insulator 



Valve 

housing 



Closed 



Seal 

liftoff 



Open 



FIGURE 6-14. The SSME main oxidizer valve is a low-pressure drop ball valve rep- 
resentative of high-pessure large valves used in rocket engines. The ball and its integral 
shaft rotate in two bearings. The seal is a machined plastic ring spring-loaded by a bel- 
lows against the inlet side of the ball. Two cams on the shaft lift the seal a short distance 
off the ball within the first few degrees of ball rotation. The ball is rotated by a preci- 
sion hydraulic actuator (not shown) through an insulating coupling. (Courtesy of Pratt & 
Whitney Rocketdyne.) 



Sudden closing of valves can cause water hammer in the pipelines, leading 
to unexpected pressure rises that can be destructive to propellant system compo- 
nents. An analysis of this water hammer phenomenon will allow determination of 
the approximate maximum pressure (Ref. 6-18). The friction of the pipe and the 
branching of pipelines reduce this maximum pressure. Water hammer can also 
occur when admitting the initial flow of high-pressure propellant into evacuated 
pipes. The pipes are under vacuum to remove air and prevent the forming of gas 
bubbles in the propellant flow, which can cause combustion problems. 

Many liquid rocket engines have filters in their lines. This is necessary to 
prevent dirt, particles, or debris, such as small pieces from burst diaphragms, from 
entering precision valves or regulators (where debris can cause a malfunction) 
or from plugging small injection holes, which could cause hot streaks in the 
combustion gases, in turn causing a thrust chamber failure. 

Occasionally, a convergent-divergent venturi section, with a sonic velocity 
at its throat, is placed into one or both of the liquid propellant lines. It has 
also been called a cavitating venturi, when the local throat pressure goes below 




6.10. ENGINE SUPPORT STRUCTURE 239 



Bearing Bellows seal 




Sleeve 



FIGURE 6-15. Flexible high-pressure joint with external gimbal rings for a high- 
pressure hot turbine exhaust gas. (Courtesy of Pratt & Whitney Rocketdyne) 



the vapor pressure. The merits are that it maintains constant flow and prevents 
pressure disturbances from traveling upstream. This can prevent the propagating 
of chamber pressure oscillations or coupling with thrust chamber combustion 
instabilities. The venturi section can also help in minimizing some water hammer 
effects in a system with multiple banks of thrust chambers. 



6.10. ENGINE SUPPORT STRUCTURE 

Most of the larger rocket engines have their own mounting structure or support 
structure. On it the major components are mounted. It also transmits the thrust 
force to the vehicle. Welded tube structures or metal plate/sheet metal assemblies 
have been used. In some large engines the thrust chamber is used as a structure 
and the turbopump, control boxes, or gimbal actuators are attached to it. 

In addition to the thrust load, an engine structure has to withstand forces 
imposed by vehicle maneuvers (in some cases a side acceleration of 10 go), 
vibration forces, actuator forces for thrust vector control motions, and loads from 
transportation over rough roads. 

In low-thrust engines with multiple thrusters there often is no separate engine 
mounting structure; the major components are in different locations of the vehicle, 
connected by tubing, wiring, or piping, and each is usually mounted directly to 
the vehicle or spacecraft structure. 



240 LIQUID PROPELLANT ROCKET ENGINE EUNDAMENTALS 



PROBLEMS 



1 . In an engine with a gas generator engine cycle, the turbopump has to do more work in 
the pumps, if the thrust chamber operating pressure is raised. This of course requires 
an increase in turbine gas flow which, when exhausted, adds little to the engine specific 
impulse. If the chamber pressure is raised too much, the decrease in performance due 
to an excessive portion of the total propellant flow being sent through the turbine and 
the increased mass of the turbopump will outweigh the gain in specific impulse that 
can be attained by increased chamber pressure and also by increased thrust chamber 
nozzle exit area. Outline in detail a method for determining the optimum chamber 
pressure where the sea-level performance will be a maximum for a rocket engine that 
operates in principle like the one shown in Fig. 1 -4. 

2 . The engine performance data for a turbopump rocket engine system are as follows: 



Propellants 

Engine system specific impulse (steady state) 

Engine system mixture ratio 

Rated engine system thrust 

Oxidizer vapor flow to pressurize oxidizer 

tank 

Propellant flow through turbine at rated thrust 
Gas generator mixture ratio 
Specific impulse of turbine exhaust 



Liquid oxygen/kerosene 
272 sec 
2.52 

40,000 N 

0.003% of total oxidizer flow 

2.1% of total propellant flow 
0.23 
85 sec 



Determine performance of the thrust chamber L,r,F (see Section 11.2). 

3 . For a pulsing rocket engine, assume a simplified parabolic pressure rise of 0.005 sec, 
a steady-state short period of full chamber pressure, and a parabolic decay of 0.007 
sec approximately as shown in the sketch. Plot curves of the following ratios as a 
function of operating time t from t = 0.013 to t = 0.200 sec: (a) average pressure 
to ideal steady-state pressure (with zero rise or decay time); (b) average L to ideal 
steady-state L; (c) average F to ideal steady-state F . 




Time 



PROBLEMS 241 



4 . For a total impulse of 100 Ibf-sec compare the volume and approximate system weights 
of a pulsed propulsion system using different gaseous propellants, each with a single 
spherical gas storage tank (at 3500 psi and 0°C). A package of small thrust nozzles with 
piping, valves, and controls is provided which weighs 3.2 lb. The gaseous propellants 
are hydrogen, nitrogen, or argon (see Table 7-3). 

5. Compare several systems for a potential roll control application which requires four 
thrusters of 1 Ibf each to operate for a cumulative duration of 2 min each over a period 
of several days, which allows a constant gas temperature. Include the following: 

Pressurized helium 70°F temperature 

Pressurized nitrogen 70°F Ambient temperature 

Pressurized krypton 70°F Ambient temperature 

Pressurized helium at 300°F (electrically heated) 



The pressurized gas is stored at 5000 psi in a single spherical fiber-reinforced plastic 
tank; use a tensile strength of 150,000 psi and a density of 0.050 Ibm/in.^ with a 
0.012-in. -thick aluminum inner liner as a seal against leaks. Neglect the gas volume 
in the pipes, valves, and thrusters, but assume the total hardware mass of these to be 
about 1.3 Ibm. Use Table 7-3. Make estimates of the tank volume and total system 
weight. Discuss the relative merits of these systems. 

6. A sealed propellant tank contains hydrazine. It is stored for long periods of time, 
and therefore the propellant and the tank will reach thermal equilibrium with the 
environment. At an ambient temperature of 20°C and an internal pressure of 1.2 atm 
the liquid occupies 87% of the tank volume and the helium pressurizing gas occupies 
13%. Assume no evaporation of the propellant, no dissolving of the gas in the liquid, 
and no movement of the tank. Use the hydrazine properties from Figs. 7-1 and 7-2 
and Table 7-1. What will be the approximate volume percentages and the gas pressure 
at the extreme storage temperatures of 4 and 40° C? 

7. A liquid hydrogen/liquid oxygen thrust chamber has a constant bipropellant flow rate 
of 347 kg/sec at a mixture ratio of 6.0. It operates at full thrust for exactly 2 min. The 
propellant in the tanks are initially at the propellant boiling points and are assumed to 
be of uniform initial temperature at start. Use data from Table 7-1 for the propellant 
specific gravities. Assuming no losses, find the masses of (a) fuel and (b) of oxidizer 
used to produce the thrust for the nominal duration, (c) What was the volume of 
the liquid hydrogen actually used? (d) Assuming 4.0% extra fuel mass (for unusable 
propellant residual, evaporation, hardware cooling, or venting just prior to start, or 
propellant consumed inefficiently during startup and shutdown) and a 10% ullage 
volume (the void space above the liquid in the tank), what will be the volume of the 
fuel tank? Assume other losses can be neglected. 

Answers: (a) 5941 kg, (b) 35,691 kg, (c) 83.83 m^ (d) 95.6 m^. 

8. Prepare dimensioned rough sketches of the four propellant tanks needed for operating 
a single gimbal-mounted RD 253 engine (Table 11-3) for 80 sec at full thrust and 



242 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



an auxiliary rocket system with a separate pressurized feed system using the same 
propellants, with two gimbal-mounted small thrust chambers, each of 150 kg thrust, a 
duty cycle of 12% (fires only 12% of the time), but for a total flight time of 1.00 hr. 
For propellant properties Table 7-1. Describe any assumptions that were made with 
the propellant budget, the engines, or the vehicle design, as they affect the amount of 
propellant. 

9. Table 10-5 shows that the RD 120 rocket engine can operate at thrusts as low as 85% 
of full thrust and with a mixture ratio variation of ±10.0%. Assume a 1.0% unavailable 
residual propellant. The allowance for operational factors, loading uncertainties, off- 
nominal rocket performance, and a contingency is 1.27% for the fuel and 1.15% for 
the oxidizer. 

(a) In a particular flight the average main thrust was 98.0% of nominal and the mixture 
ratio was off by ±2.00% (oxidizer rich). What percent of the total fuel and oxidizer 
loaded into the vehicle will remain unused at thrust termination? 

(b) If we want to run at a fuel-rich mixture in the last 20% of the flight duration (in 
order to use up all the intended flight propellant), what would the mixture ratio 
have to be for this last period? 

(c) In the worst possible scenario with maximum throttling and extreme mixture ratio 
excursion (±3.00%, but operating for the nominal duration), what is the largest 
possible amount of unused oxidizer or unused fuel in the tanks? 



SYMBOLS 



m 

m 

P 

Ap 



R 

T 

V 



w 

w 



Cf 

F 

go 

Is 

k 



C 



r 



effective exhaust velocity, m/sec (ft/sec) 
specific heat at constant volume, J/kg-K (Btu/lbm-°R) 
thrust coefficient 
thrust force, N (Ibf) 

acceleration of gravity at sea level, 9.8066 m/sec^ 

specific impulse, sec 

specific heat ratio 

propellant mass, kg (Ibm) 

mass flow rate, kg/sec (Ib/sec) 

pressure, N/m^ (psi) 

pressure difference, N/m^ (psi) 

mixture ratio (oxidizer to fuel mass flow rates) 

gas constant per unit mass, J/kg-K (ft-lbf/lbm-°R) 

absolute temperature, K 

volume flow rate, m^/sec (ft^/sec) 

total propellant weight, N (Ibf) 

weight flow rate, N/sec (Ibf/sec) 



REFERENCES 243 



Subscripts 



/ 

0 

g 



fuel 



initial condition or stagnation condition 

gas tank 

oxidizer 

propellant tank or power cutoff 



o 



P 



REFERENCES 

6-1. D. K. Huzel and D. H. Huang, Design of Liquid Propellant Rocket Engines, rev. 
ed., AIAA, Reston, VA, 1992. 

6-2. G. G. Gakhun, V. I. Baulin, et al. Construction and Design of Liquid Propel- 
lant Rocket Engines (in Russian), Konstruksiya i Proyektirovaniye Zhidkostniyk 
Raketnykh Dvigateley , Mashinostroyeniye, Moscow, 1989. 

6-3. J. I. Hochsten, H.-C. Ji, and J. Ayelott, “Prediction of Self-Pressurization Rate of 
Cryogenic Propellant Tankage,” Journal of Propulsion and Power, Vol. 6, No. 1, 
January-February 1990, pp. 11-17. 

6-4. B. Morton, M. Elgersma, and R. Playter, “Analysis of Booster Vehicle Slosh Sta- 
bility During Ascent to Orbit,” AIAA Paper 90-1876, July 1990. 

6-5. J. J. Pocha, “Propellant Slosh in Spacecraft and How to Live with It,” Aerospace 
Dynamics, Vol. 20, Autumn 1986, pp. 26-31. 

6-6. G. P. Purohit and L. D. Loudenback, “Application of Etched Disk Stacks in Surface 
Tension Propellant Management Devices,” Journal of Propulsion and Power, Vol. 
7, No. 1, January-February 1991, pp. 22-30. 

6-7. J. R. Rollins, R. K. Grove, and D. R. Walling, Jr. “Design and Qualification of 
a Surface Tension Propellant Tank for an Advanced Spacecraft,” AIAA Paper 
88-2848, 24th Joint Propulsion Conference, 1988. 

6-8. Design Guide for Pressurized Gas Systems, Vols. I and II, prepared by JIT Research 
Institute, NASA Contract NAS7-388, March 1966. 

6-9. H. C. Hearn, “Evaluation of Bipropellant Pressurization Concepts for Spacecraft,” 
Journal of Spacecraft and Rockets, Vol. 19, July 1982, pp. 320-325. 

6-10. H. C. Hearn, “Design and Development of a Large Bipropellant Blow- 
down Propulsion System,” Journal of Propulsion and Power, Vol. 11, No. 5, 
September-October 1995. 

6-11. National Space Transportation System Reference, Vol. 1, National Aeronautics and 
Space Administration, Washington, DC, June 1988 (description of Space Shuttle 
system and operation). 

6-12. Personal communications from Jim Morehart, The Aerospace Corp., Vince Whee- 
lock, Pat Mills, and Randy Parsley of Pratt & Whitney Rocketdyne, and Gordon 
Dressier, Northrop Grumman Corp. 



244 LIQUID PROPELLANT ROCKET ENGINE FUNDAMENTALS 



6-13. H. Grosdemange and G. Schaeffer, “The SEPR 844 Reuseable Liquid Rocket 
Engine for Mirage Combat Aircraft,” AIAA Paper 90-1835, July 1990. 

6-14. D. Manski, C. Goertz, H. D. Sassnick, J. R. Hulka, B. D. Goracke, and D. J. H. 
Levack, “Cycles for Earth to Orbit Propulsion,” Journal of Propulsion and Power, 
AIAA, Vol. 14, No. 5, September- October 1998. 

6-15. D. Conley, N. Y. Lee, P. L. Portanova and B. K. Wood, “Evolved Expendable 
Launch Vehicle System: RS-68 Main Engine Development,” Paper 1AC-02.S.1.01, 
53rd International Astronautical Congress, 10-19 Oct. 2002, Houston, Texas. 

6-16. J. R. Brown, “Expander Cycle Engines for Shuttle Cryogenic Upper Stages,” 
AIAA Paper 83-1311, 1983. 

6-17. G. P. Sutton, “History of Small Liquid Propellant Thrusters,” presented at 52nd 
JANNAF Propulsion Meeting, 12 May 2004, Las Vegas, Nevada; published by 
Chemical Propulsion Information Analysis Center, Columbia, MD. 

6-18. R. P. Prickett, E. Mayer, and J. Hermel, “Waterhammer in Spacecraft Propellant 
Feed Systems,” Journal of Propulsion and Power, Vol. 8, No. 3, May-June 1992. 



CHAPTER 7 



LIQUID PROPELLANTS 



Section 6.1 gave the classification of liquid propellants. In this chapter we 
discuss properties, performance, hazards, and characteristics of selected common 
liquid propellants. These characteristics affect the engine and vehicle design, test 
facilities, and propellant storage and handling. Today we commonly use three 
liquid bipropellant combinations. Each of these propellants will be described 
further in this chapter. They are: (1) the cryogenic oxygen-hydrogen propel- 
lant system, used in upper stages and sometimes booster stages of space launch 
vehicles; it gives the highest specific impulse for a nontoxic propellant combi- 
nation, which makes it best for high vehicle velocity missions; (2) the liquid 
oxygen-hydrocarbon propellant combination, used for booster stages (and a few 
second stages) of space launch vehicles; its higher average density allows a more 
compact booster stage, when compared to the first combination; also, historically, 
it was developed before the first combination and was originally used for ballistic 
missiles; (3) not a single propellant combination, but several storable propellant 
combinations, are used in large rocket engines for first and second stages of 
ballistic missiles and in almost all bipropellant low-thrust, auxiliary or reaction 
control rocket engines (this term is defined below); they allow long-term storage 
and almost instant readiness to start without the delays and precautions that come 
with cryogenic propellants. There also were fuel mixtures of several storable 
propellant such as alcohol and hydrazine. In Russia the nitric acid-hydrocarbon 
combination was used in ballistic missiles many years ago. Today Russia and 
China favor nitrogen tetroxide as the oxidizer-unsymmetrical dimethylhydrazine, 
or UDMH, as the fuel for ballistic missiles and auxiliary engines. The United 
States started with nitrogen tetroxide and a fuel mixture of 50% UDMH with 50% 
hydrazine in the large engines for the Titan II missile. For auxiliary engines in 
many satellites and upper stages the United States has used, and is still using, the 



245 



246 LIQUID PROPELLANTS 



bipropellant of nitrogen tetroxide with monomethylhydrazine. The orhit maneu- 
vering system of the Space Shuttle uses it. Alternatively many U.S. satellites 
have used, and are still using monopropellant hydrazine for low-thrust auxiliary 
engines. 

In the last 30 years there has not been a truly new propellant used in a 
rocket engine for an operational flight vehicle. Some new propellants, such as 
hydroxyl ammonium nitrate, have been synthesized, manufactured, and ground 
tested in thrust chambers in the last decade, but they have not found their way 
into an operational rocket engine. Between 1942 and 1975 a number of other 
propellants (not the ones commonly used today) were successfully flown; this 
includes ammonia (X-15 Research test aircraft), ethyl alcohol (German V-2 or 
U.S. Redstone), or aniline (WAC Corporal). They each had some disadvantages 
and they are no longer used for operational flights today. The liquid oxidizer 
nitrous oxide (N 2 O), also known as laughing gas, has found a new application 
in a hybrid propulsion system (with a solid fuel) and it is discussed further in 
Chapter 16 on hybrid propulsion systems. Most of the commonly used liquid 
oxidizers for liquid propellant rocket engines are listed and described briefly in 
this chapter. 

A comparative listing of various performance quantities for a number of pro- 
pellant combinations is given in Table 5-5 and in Ref. 7-1. Some important 
physical properties of selected common propellants are given in Table 7-1. For 
comparison water is also listed. Specific gravities and vapor pressures are shown 
in Figs. 7-1 and 7-2. 



7.1. PROPELLANT PROPERTIES 

It is important to distinguish between the characteristics and properties of the 
liquid propellants (the fuel and oxidizer liquids in their unreacted condition) and 
those of the hot gas mixture, which results from the reaction in the combustion 
chamber. The chemical nature of the liquid propellants and their mixture ratio 
determines the properties and characteristics of both of these types. Unfortunately, 
none of the practical, known propellants have all the desirable properties, and 
the selection of the propellant combination is usually a compromise of various 
factors, such as those listed below. 

Economic Factors 

Availability in quantity and a low cost are very important considerations in the 
selection of a propellant. In military applications, consideration has to be given to 
logistics of production, supply, storage, and other factors. The production process 
should be simple, requiring only ordinary chemical equipment and available raw 
materials. It is usually more expensive to use a toxic or cryogenic propellant than 
a storable, nontoxic one, because it requires additional steps in the operation, more 
safety provisions, additional design features, longer check-out procedures prior 
to launch, and often better trained personnel. 



Temperature, K 

100 150 200 250 300 350 




o 

o 

eg 

+ 



O 




o 

o 

CM 



O 

O 

ro 

I 



o 

o 



247 



FIGURE 7-1. Specific gravities of several liquid propellants as a function of temperature. 



248 LIQUID PROPELLANTS 



TABLE 7-1. Physical Properties of Several Common Liquid Propellants 



Propellant 


Liquid 

Fluorine 


Hydrazine 


Liquid 

Hydrogen 


Methane 


Monomethyl- 

hydrazine 


Chemical formula 


P2 


N 2 H 4 


H 2 


CH 4 


CH 3 NHNH 2 


Molecular mass 


37.997 


32.045 


2.016 


16.03 


46.072 


Melting or freezing 


53.54 


275.16 


14.0 


90.5 


220.7 


point (K) 


Boiling point (K) 


85.02 


387.46 


20.4 


111.6 


360.8 


Heat of vaporization 


172.2'' 


1219'' 


446 


510'' 


808 


(kJ/kg) 


Specific heat 


0.368 


0.736 


1.75'' 


0.835'' 


0.700 


(kcal/kg-K) 


(85 K) 


(293 K) 


(20.4 K) 




(298 K) 




0.357 


0.758 


— 


— 


0.735 


Specific gravity" 


(69.3 K) 
1.636 


(338 K) 
1.0037 


0.071 


0.424 


(393 K) 
0.8702 




(66 K) 


(298 K) 


(20.4 K) 


(111.5 K) 


(298 K) 




1.440 


0.952 


0.076 


— 


0.857 


Viscosity 


(93 K) 
0.305 


(350 K) 
0.97 


(14 K) 
0.024 


0.12 


(311 K) 
0.775 


(centipoise) 


(77.6 K) 


(298 K) 


(14.3 K) 


(111.6 K) 


(298 K) 




0.397 


0.913 


0.013 


0.22 


0.40 




(70 K) 


(330 K) 


(20.4 K) 


(90.5 K) 


(344 K) 


Vapor pressure 


0.037 


0.0019 


0.2026 


0.033 


0.0066 


(MPa) 


(77 K) 


(298 K) 


(23 K) 


(100 K) 


(298 K) 




0.0065 


0.016 


0.87 


0.101 


0.638 




(66.5 K) 


(340 K) 


(30 K) 


(117 K) 


(428 K) 



"Red fuming nitric acid (RFNA) has 5 to 20% dissolved NO 2 with an average molecular mass of 
about 60, and a density and vapor pressure somewhat higher than those of pure nitric acid. 

*At boiling point. 

"Reference for specific gravity ratio: 10^ kg/m^ or 62.42 Ibm/ft^. 



Performance of Propellants 

The rocket engine performance can be compared on the basis of the specific 
impulse, the exhaust velocity, the characteristic velocity, or other engine param- 
eters. They have been explained in Chapters 3, 5, and 6. The specific impulse 
and exhaust velocity are functions of pressure ratio, specific heat ratio, com- 
bustion temperature, mixture ratio, and molecular mass. Values of performance 
parameters for various propellant combinations can be calculated with a high 
degree of accuracy and several are listed in Table 5-5. Very often the perfor- 
mance is expressed in terms of fiight performance parameters for a given rocket 



7.1. PROPELLANT PROPERTIES 249 



Nitric Acid" 
(99% pure) 


Nitrogen 

Tetroxide 


Liquid 

Oxygen 


Rocket Fuel 
RP-1, RP-2 


Unsymmetrical 

Dimethyl- 

hydrazine 

(UDMH) 


Water 


HNO 3 


N 2 O 4 


O 2 


Hydrocarbon 

CH 1.97 


(CH3)2NNH2 


H 2 O 


63.016 


92.016 


31.988 


~ 175 


60.099 


18.02 


231.6 


261.95 


54.4 


225 


216 


273.15 


355.7 


294.3 


90.0 


460-540 


335.5 


373.15 


480 


413* 


213 


246* 


543 


2253* 


0.042 


0.374 


0.4 


0.45 


0.704 


1.008 


(311 K) 
0.163 
(373 K) 


(290 K) 
0.447 
(360 K) 


(65 K) 


(298 K) 


(298 K) 
0.715 
(340 K) 


(273.15 K) 


1.549 


1.447 


1.14 


0.58 


0.7861 


1.002 


(273.15 K) 


(293 K) 


(90.4 K) 


(422 K) 


(298 K) 


(373.15 K) 


1.476 


1.38 


1.23 


0.807 


0.784 


1.00 


(313.15 K) 


(322 K) 


(77.6 K) 


(289 K) 


(244 K) 


(293.4 K) 


1.45 


0.47 


0.87 


0.75 


0.492 


0.284 


(273 K) 


(293 K) 


(53.7 K) 


(289 K) 


(298 K) 


(373.15 K) 




0.33 


0.19 


0.21 


0.48 


1.000 




(315 K) 


(90.4 K) 


(366 K) 


(300 K) 


(277 K) 


0.0027 


0.01014 


0.0052 


0.002 


0.0223 


0.00689 


(273.15 K) 


(293 K) 


(88.7 K) 


(344 K) 


(298 K) 


(312 K) 


0.605 


0.2013 




0.023 


0.1093 


0.03447 


(343 K) 


(328 K) 




(422 K) 


(339 K) 


(345 K) 



application, as explained in Chapter 4. Here the average density, the total impulse, 
and the engine mass ratio usually enter into a complex flight relation equation. 

For high performance a high content of chemical energy per unit of propel- 
lant mixture is desirable because it permits a high chamber temperature. A low 
molecular mass of the product gases of the propellant combination is also desir- 
able. It can be accomplished by using fuels rich in combined hydrogen, which is 
liberated to become gas during the reaction. A low molecular mass is obtained if 
a large portion of the hydrogen gas produced does not combine with oxygen. In 
general, therefore, the best mixture ratio for many bipropellants is not necessar- 
ily the stoichiometric one (which results in complete oxidation and yields a high 
flame temperature) but usually a fuel-rich mixture containing a large portion of 
low-molecular-mass reaction products, as shown in Chapter 5. 



Vapor pressure, atm 



Temperature, K 

0 100 200 300 400 500 




250 



FIGURE 7-2. Vapor pressures of several liquid propellants as a function of temperature. 



7.1. PROPELLANT PROPERTIES 251 



If very small metallic fuel particles of beryllium or aluminum are suspended 
in the liquid fuel, it is theoretically possible to increase the specific impulse 
by between 9 and 18%, depending on the particular propellant combination, its 
mixture ratio, and the metal powder additive. Gelled propellants with suspended 
small solid particles have been ground tested successfully with storable fuels. 
For gelled propellants, see Section 7.5. 

The chemical propellant combination that has the highest potential specific 
impulse (approximately 480 sec at 1000 psia chamber pressure and expansion 
to sea-level atmosphere, and 565 sec in a vacuum with a nozzle area ratio of 
50) uses a toxic liquid fluorine oxidizer with hydrogen fuel plus suspended toxic 
solid particles of beryllium; as yet a practical means for storing these propellants 
and a practical rocket engine have not been developed. 

Common Physical Hazards 

Although the several categories of hazards are described below, they do not all 
apply to every propellant or every bipropellant combination. The hazards are 
different for each specific propellant and must be carefully understood before 
working with that propellant. The consequences of unsafe operation or unsafe 
design are usually also unique to several propellants. 

Corrosion. Various propellants, such as nitrogen tetroxide, nitric acid, or hydro- 
gen peroxide, have to be handled in containers and pipelines of special materials. 
If the propellant were permitted to become contaminated with corrosion prod- 
ucts, its physical and chemical properties could change sufficiently to make it 
unsuitable for rocket operation. The corrosion of the gaseous reaction products 
is important in applications in which the reaction products are likely to damage 
structure and parts of the vehicle or affect communities and housing near a test 
facility or launch site. 

Explosion Hazard. Some propellants, such as hydrogen peroxide and 
nitromethane, are unstable and tend to detonate under certain conditions of impu- 
rities, temperature, and shock. If liquid oxidizers (e.g., liquid oxygen) and fuels 
are mixed together, they can be detonated. Unusual, rare flight vehicle launch or 
transport accidents have caused such mixing to occur (see Refs. 7-2 and 7-3). 

Fire Hazard. Many oxidizers will start chemical reactions with a large variety of 
organic compounds. Nitric acid, nitrogen tetroxide, fluorine, or hydrogen peroxide 
react spontaneously with many organic substances, resulting in a conflagration. 
Most of the fuels are readily ignitable when exposed to air and heat. Also some 
household dusts, certain paints, or smoke particles can be oxidized. 

Accidental Spills. Unforeseen mishaps during engine operation and traffic acci- 
dents on highways or railroads while transporting hazardous materials, including 
propellants, have on occasion caused spills, which expose people to unexpected 



252 LIQUID PROPELLANTS 



fires or potential health hazards. The U.S. Department of Transportation has 
rules for marking and containing hazardous materials during transport and also 
guidelines for emergency action (see Ref. 7-4). 

Health Hazards. Many propellants are toxic or poisonous, and special pre- 
cautions have to be taken to protect personnel. Fluorine, for example, is very 
poisonous. Toxic propellant chemicals or poisonous exhaust species can enter 
the human body in several ways. The resulting health disorders are propellant 
specific. Nitric acid can cause severe skin burns and tissue disintegration. Skin 
contact with aniline or hydrazine can cause nausea and other adverse health 
effects. Hydrazine, monomethylhydrazine, unsymmetrical dimethylhydrazine, or 
hydrazine hydrate are known animal carcinogens (cancer-causing substances) 
and suspected human carcinogens. Many propellant vapors cause eye irritation, 
even in very small concentration. Inadvertent ingestion (swallowing) of many 
propellants can also cause severe health degradation. 

The inhalation of certain toxic exhaust gases or gaseous or vaporized propel- 
lants is perhaps the most common health hazard. It can cause severe damage 
if the exposure is for long duration or in concentrations that exceed established 
maximum threshold values. In the United States the Occupational Safety and 
Health Administration (OSHA) has established limits or thresholds on the allow- 
able exposure and concentration for most propellant chemicals. Several of these 
threshold limits are mentioned later in this chapter. Toxic gases in the exhaust 
could include hydrofluoric acid (HF) gas; its OSHA 8-hr personnel exposure 
limit is 3 ppm (volumetric parts per million) and its short-term (typically, 15 
min) exposure limit is 6 ppm. A concentration of 3000 ppm or 0.3% can be fatal 
within a few seconds. References 7-2 and 7-5 give more information on toxic 
effects. 

The corrosion, explosion, and fire hazards of many propellants put severe 
limitations on the materials, the handling, and the design of rocket-propelled 
vehicles and their vehicle’s engine compartments. Not only is the rocket system 
itself exposed to the hazardous propellant, but adjacent personnel, structural parts, 
electrical and other vehicle equipment, and test and launch facilities have to 
be properly protected against the effects of possible leaks, fumes, and fires or 
explosions from propellant accumulations. 

Material Compatibility. Several liquid propellants have only a limited num- 
ber of truly compatible materials, both metals and nonmetals, such as gaskets or 
0-rings. There have been unfortunate failures (causing fires, leakage, corrosion, 
or malfunctions) when an improper or incompatible material was used in the 
hardware of a rocket engine. Depending on the specific component and load- 
ing conditions, these structural materials have to withstand high stresses, stress 
corrosion, and in some versions high temperatures or abrasion. Several specific 
material limitations are mentioned in the next section. Certain materials catalyze 
a self-decomposition of stored hydrogen peroxide into water and oxygen, mak- 
ing long-term storage difficult and, if confined, causing its closed container to 



7.1. PROPELLANT PROPERTIES 253 



explode. Many structural materials, when exposed to cold, cryogenic propellants, 
can become very brittle. 



Desirable Physical Properties 

Low Freezing Point. This permits operation of rockets in cold weather. The 
addition of small amounts of freezing point depressant has been found to help 
lower the freezing point of some liquid propellants which otherwise might solidify 
at relatively high temperature. 



High Specific Gravity. A denser propellant will provide a larger mass of pro- 
pellants in a given vehicle tank volume. It permits a smaller tank volume and, 
consequently, a lower structural vehicle mass and lower aerodynamic drag. Pro- 
pellant specific gravity, therefore, has an important effect on the maximum flight 
velocity and range of any rocket-powered vehicle or missile, particularly when 
flying within the Earth’s atmosphere, as explained in Chapter 4. Specific gravities 
for various propellants are plotted in Fig. 7-1. A variation of the temperature of 
stored propellant will cause a change of the liquid level in the tank. 

For any given mixture ratio r, the average specific gravity of a propellant 
combination Sav can be determined from the specific gravities of the fuel Sf and of 
the oxidizer So ■ The average specific gravity is defined as the mass of the fuel and 
oxidizer, divided by the sum of their volumes. Here the mixture ratio is defined 
as the oxidizer mass flow rate divided by the fuel mass flow rate. (Eq. 6-1): 



^ So8f(l+r) 

rSf + So 



(7-1) 



Values of 5av for various propellant combinations are listed in Table 5-5. The 
value of 5av can be increased by adding heavy materials to the propellants, either 
by solution or colloidal suspension. The identical type of equation can be written 
for the average density pav in terms of the fuel density and the oxidizer density: 



Pav — 



PoPfjl + r) 
Pfr + Po 



(7-2) 



In the SI system of units the specific gravity has the same numerical value 
as the density expressed in units of grams per cubic centimeter or kg/liter. In 
some performance comparisons the parameter density specific impulse Id is used. 
It is defined as the product of the average specific gravity S and the specific 
impulse 4: 



Id = &..Is (7-3) 

Stability. No deterioration and no decomposition with long-term (over 15 years) 
storage and minimal reaction with the atmosphere have been attained with many 
propellants. Good chemical stability means no decomposition of the liquid pro- 
pellant during operation or storage, even at elevated temperature. A good liquid 



254 LIQUID PROPELLANTS 



propellant should also have no chemical deterioration when in contact with tub- 
ing pipes, tank walls, valve seats, and gasket materials, even at relatively high 
ambient temperatures. No appreciable absorption of moisture and no adverse 
effects of small amounts of impurities are desirable properties. There should be 
no chemical deterioration when liquid flows through the hot cooling jacket pas- 
sages of a regeneratively cooled thrust chamber. Some hydrocarbons (e.g., some 
olefins) decompose and form carbonaceous deposits on the hot inside surfaces 
of the cooling passage. These deposits can become hard, reduce the heat flow, 
increase the local metal temperatures, and thus can cause the metal to weaken and 
fail. About 1% per year of stored 90% concentrated hydrogen peroxide decom- 
poses even in clean storage tanks. Between 1 and 20% of a cryogenic propellant 
(stored in a flight vehicle) evaporates every day in an insulated tank. 

Heat Transfer Properties. High specific heat, high thermal conductivity, and 
a high boiling or decomposition temperature are desirable for propellants that are 
used for thrust chamber cooling (see Section 8.5). 

Pumping Properties. A low vapor pressure permits not only easier handling 
of the propellants, but also a more effective pump design in applications where 
the propellant is pumped. This reduces the potential for cavitation, as explained in 
Chapter 10. If the viscosity of the propellant is too high, then pumping and engine- 
system calibration become difficult. Propellants with high vapor pressure, such as 
liquid oxygen, liquid hydrogen, and other liquefied gases, require special design 
provisions, unusual handling techniques, and special low-temperature materials. 

Temperature Variation of Physical Properties. The temperature variation 
of the physical properties of the liquid propellant should be small and should 
be very similar for the fuel and for the oxidizer. For example, a wide tempera- 
ture variation in vapor pressure and density (thermal coefficient of expansion) or 
an unduly high change in viscosity with temperature makes it difficult to accu- 
rately calibrate a rocket engine flow system or predict its performance over any 
reasonable range of operating temperatures. 

Ignition, Combustion, and Flame Properties 

If the propellant combination is spontaneously ignitable, it does not require an 
ignition system. This means that burning is initiated as soon as the oxidizer and 
the fuel come in contact with each other. Spontaneously or self-ignitable propel- 
lant combinations are often termed hypergolic propellants. Although an ignition 
system is not a very objectionable feature, its elimination is usually desirable 
because it simplifies the propulsion system. All rocket propellants should be read- 
ily ignitable and have only a very short ignition time delay in order to reduce 
the potential explosion hazard during starting. Starting and ignition problems are 
discussed further in Section 8.6. 

Nonspontaneously ignitable propellants have to be heated by external means 
before ignition can begin. Igniters are devices that accomplish an initial slight 



7.1. PROPELLANT PROPERTIES 255 



pressurization of the chamber and the initial heating of the propellant mixture 
to the point where steady flow combustion can be self-sustained. The amount 
of energy added by the igniter to activate the propellants should be small so 
that low-power ignition systems can be used. The energy required for satis- 
factory ignition usually diminishes for increasing ambient temperature of the 
propellant. At low ambient temperatures ignition can be very slow (0.05 to 
0.02 sec). 

Certain propellant combinations bum very smoothly without combustion vibra- 
tion which are gas pressure oscillations. Other propellant combinations do not 
demonstrate this combustion stability and, therefore, are less desirable. Combus- 
tion is treated in Chapter 9. 

Smoke formation is objectionable in many applications because of the smoke 
deposits on the surrounding equipment and parts. Smoke and brilliantly luminous 
exhaust flames are objectionable in certain military applications because they 
can be easily detected. In some applications the condensed species in the exhaust 
gas can cause surface contamination on spacecraft windows or optical lenses, 
and the electrons in the flame can cause undesirable interference or attenuation 
of communications radio signals . See Chapter 20 for information on exhaust 
plumes. 



Property Variations and Specifications 

The propellant properties and quality must not vary from batch to batch, because 
this can affect engine performance, combustion, and physical or chemical prop- 
erties. The same propellant must have the same composition, properties, and 
storage or rocket operating characteristics if manufactured at different times or 
if made by different manufacturers. For these reasons propellants are purchased 
against specifications which define ingredients, maximum allowable impurities, 
packaging methods or compatible materials, allowable tolerances on physical 
properties (such as density, boiling point, freezing point, viscosity, or vapor 
pressure), quality control requirements, cleaning procedures for containers, doc- 
umentation of inspections, laboratory analyses, or test results. A careful chemical 
analysis of the composition and impurities is necessary. Reference 7-6 describes 
some of these methods of analysis. 



Additives 

Altering and tailoring propellant properties can be achieved with additives. For 
example, to make a nonhypergolic fuel become hypergolic (readily ignited), a 
reactive ingredient has been added. To desensitize concentrated hydrogen perox- 
ide and reduce self-decomposition, it is diluted with 3 to 15% of clean water. 
To increase density or to alleviate certain combustion instabilities, a fine pow- 
der of a heavy solid material, such as aluminum, can be suspended in the fuel. 
The use of additives to lower the freezing point temperature has already been 
mentioned. 



256 LIQUID PROPELLANTS 



7.2. LIQUID OXIDIZERS 

The most energetic oxidizer with the highest specific impulse is liquid fluorine. 
It has a high density and is extremely corrosive and toxic. It has been tested in 
several complete rocket engines but has been abandoned because of its hazards. 
Many different types of new storable and cryogenic liquid oxidizer propellants 
have been synthesized, tested in small thrust chambers, or proposed. For high spe- 
cific impulse this includes boron-oxygen-fluorine compounds, oxygen-fluorine 
compounds, nitrogen-fluorine formulations, and fluorinated hydrocarbons; how- 
ever, they all have some undesirable characteristics and these synthetic oxidizers 
have not been proven to be practical. Oxidizer liquids that have been used in 
experimental liquid rocket engines include mixtures of liquid oxygen and liq- 
uid fluorine, oxygen difluoride (OF 2 ), chlorine trifluoride (CIF 3 ), or chlorine 
pentafluoride (CIF 5 ). All of these are highly toxic, very corrosive, and are not 
used today. Several commonly used oxidizers are listed below. 

Liquid Oxygen (O 2 ) 

Liquid oxygen, often abbreviated as LOX, boils at 90 K at atmospheric pressure; 
at these conditions it has a specific gravity of 1.14 and a heat of vaporization 
of 213 kJ/kg. It is widely used as an oxidizer and burns with a bright white- 
yellow flame with most hydrocarbon fuels. It has been used in combination 
with alcohols, jet fuels (kerosene-type), gasoline, and hydrogen. As shown in 
Table 5-5, the attainable performance is relatively high, and liquid oxygen is 
therefore a desirable and commonly used propellant in large rocket engines. The 
following missiles and space launch vehicles use oxygen: (1) with jet fuel or RP- 
1 (kerosene) — Atlas V and Soyuz (Russia); (2) with hydrogen — Space Shuttle, 
Ariane-V (France), Delta IV, and Centaur upper stage; (3) with alcohol — some 
amateur rocket engines. Figs. 1-4 and 6-1 show units that use oxygen. Figures 
5-1 to 5-6 give theoretical performance data for liquid oxygen with a kerosene- 
type fuel. 

Although it usually does not burn spontaneously with organic materials at 
ambient pressures, combustion or explosions can occur when a confined mix- 
ture of oxygen and organic matter is suddenly pressurized. Impact tests show 
that mixtures of liquid oxygen with many commercial oils or organic materials 
will detonate. Liquid oxygen supports and accelerates the combustion of other 
materials. Handling and storage are safe when contact materials are clean. Liquid 
oxygen is a noncorrosive and nontoxic liquid and will not cause the deteriora- 
tion of clean container walls. When in prolonged contact with human skin, the 
cryogenic propellant causes severe frostbite. Because liquid oxygen evaporates 
rapidly, it cannot be stored readily for any great length of time. If liquid oxygen 
is used in large quantities, it is often produced very close to its geographical 
point of application. Liquid oxygen can be obtained in several ways, such as by 
fractionated distillation of liquid nitrogen out of liquid air. 

It is necessary to insulate all lines, tanks, valves, and so on that contain liquid 
oxygen in order to reduce the evaporation loss. Rocket propulsion systems which 



7.2. LIQUID OXIDIZERS 257 



remain filled with liquid oxygen for several hours and liquid oxygen storage 
systems have to be well insulated against absorbing heat from the surroundings. 
External drainage provisions have to be made on all liquid oxygen tanks and 
lines to eliminate the water that condenses on the walls. 

Example 7-1. Estimate the approximate temperature and volume change of liquid oxy- 
gen if an oxygen tank is pressurized to 8.0 atm for a long time before engine start. Assume 
the tank is 60% full and the evaporated oxygen is caught, refrigerated, and recondensed 
(constant mass). 

SOLUTION. Using Table 7-1 and Figs. 7-1 and 7-2, the vapor pressure goes from 1.0 
atm (0.1 MPa) to 8 atm (about 0.8 MPa) and the equilibrium temperature goes from the 
boiling point of 90 K at 1.0 atm to about 133 K at 8 atm. The corresponding specific 
gravities are 1.14 and 0.88, respectively. This is an increase of 1.14/0.88 = 1.29. (29% 
more volume). The tank would be 77% full. 

In tanks with turbopump feed systems the actual tank pressures are lower (typically 2 to 
4 atm) and the evaporated oxygen is vented, causing a cooling effect on the liquid surface. 
So the numbers calculated above are too large (8 atm was selected to clearly show the 
effect). The warming occurs when there is a long hold period of a pressurized cryogenic 
propellant tank and is most pronounced when the final portion of the propellant is being 
emptied. Nevertheless the higher temperature, higher vapor pressure, and lower density 
can cause changes in mixture ratio, required tank volume, and pump suction condition 
(see Section 10.5). Therefore tanks with cryogenic propellant are insulated (to minimize 
heat transfer and density changes) and are pressurized only shortly before engine start, so 
as to keep the propellant at its lowest possible temperature and highest density. 



Hydrogen Peroxide (H2O2) 

This propellant is not only a powerful liquid oxidizer but also a clean-burning 
monopropellant. In rocket applications, hydrogen peroxide has been used in a 
highly concentrated form of 70 to 99%; the remainder is mostly water. Com- 
mercial peroxide is approximately 30% concentrated. For many paper and pulp 
mills it is 70% concentrated hydrogen peroxide. It was used in gas generators 
and rocket applications between 1938 and 1965 (X-1 and X-15 research aircraft). 
Today 90% hydrogen peroxide is the most common. 

As a monopropellant it decomposes according to the following chemical reac- 
tion, forming superheated steam and gaseous oxygen: 

H 2 O 2 ^ H 2 O + j 02 + heat 

This decomposition is brought about by the action of catalysts such as silver 
screens, various liquid permanganates, solid manganese dioxide, platinum, and 
iron oxide. In fact, most impurities act as a catalyst. H 2 O 2 is hypergolic with 
hydrazine and will burn well with kerosene. The theoretical specific impulse of 
90% hydrogen peroxide is 154 sec, when used as a monopropellant with a solid 
catalyst bed. 



258 LIQUID PROPELLANTS 



Even under favorable conditions H2O2 will often decompose at a slow rate 
during storage, about 1% per year for 90 to 95%, and gas will bubble out of the 
liquid. Contaminated liquid peroxide must be disposed of before it reaches about 
448 K, when an explosion may occur. Concentrated peroxide causes severe bums 
when in contact with human skin and may ignite and cause fires when in contact 
with wood, oils, and many other organic materials. In the past rocket engines 
with hydrogen peroxide as the oxidizer have been used for aircraft boost (German 
Me 163, and U.S. F 104) and a satellite launcher (Britain: Black Arrow). As a 
monopropellant it was used for attitude control thmsters and for gas generators. 
It has not been used for a long time, partly because of its long-term storage 
stability. However, there has been some improvement and some renewed interest 
in this dense oxidizer, which produces a nontoxic exhaust. 

Nitric Acid (HNO3) 

There are several types of nitric acid mixtures that have been used as oxidiz- 
ers between 1940 and 1965; they are not used extensively today in the United 
States. The most common type, red fuming nitric acid (RENA), consists of con- 
centrated nitric acid (HNO3) that contains between 5 and 27% dissolved nitrogen 
dioxide (NO2). The evaporating red-brown fumes are exceedingly annoying and 
poisonous. Compared to concentrated nitric acid (also called white fuming nitric 
acid), RENA is more energetic, more stable in storage, and slightly less corrosive 
to many tank materials. 

Nitric acid is highly corrosive. Only certain types of stainless steel, gold, and 
a few other materials are satisfactory as storage containers or tubing materi- 
als. A small addition of fluoride ion (less than 1% of hydrofluoric acid, or HE) 
inhibits the nitric acid, causes a protective fluoride layer to form on the wall, and 
reduces the corrosion with many metals. It is called inhibited red fuming nitric 
acid (IRENA). Even with an inhibitor, nitric acid reacts with the wall materials 
and forms dissolved nitrates and sometime insoluble nitrates. This changes the 
properties of the acid and may cause blocking of valve and injector orifices. In 
case of accidental spilling, the acid should be diluted with water or chemically 
deactivated. Lime and alkali metal hydroxides and carbonates are common neu- 
tralizing agents. However, nitrates formed by the neutralization are also oxidizing 
agents and must be handled accordingly. 

Nitric acid has been used with gasoline, various amines, hydrazine, dimethyl- 
hydrazine, and alcohols. It ignites spontaneously with hydrazine, furfuryl alcohol, 
aniline, and other amines. The specific gravity of nitric acid varies from 1.5 to 
1.6, depending on the percentages of nitrogen dioxide, water, and impurities. 
This high density permits compact vehicle construction. 

Vapors from nitric acid or red fuming nitric acid have an OSHA 8-hr personnel 
exposure limit or a threshold work allowance of 2 ppm (parts per million or about 
5 mg/m^) and a short-term exposure limit of 4 ppm. Droplets on the skin cause 
burns and sores which do not heal readily. 



7.3. LIQUID FUELS 259 



Nitrogen Tetroxide (N2O4), Often Abbreviated as NTO 

This is a high-density yellow-brown liquid (specific gravity of 1.44). It is com- 
monly used with the hypergolic fuels hydrazine, monomethylhydrazine (MMH), 
and unsymmetrical dimethylhydrazine (UDMH). Although it is the most com- 
mon storable oxidizer today, its liquid temperature range is narrow and it is easily 
frozen or vaporized. It is only mildly corrosive when pure, but forms strong acids 
when moist or allowed to mix with water. It readily absorbs moisture from the 
air. It can be stored indefinitely in sealed containers made of compatible material. 
It is hypergolic with many fuels and can cause spontaneous ignition with many 
common materials, such as paper, grease, leather, or wood. The NO2 fumes are 
reddish brown and are extremely toxic. Because of its high vapor pressure it must 
be kept in relatively heavy tanks. The freezing point of N2O4 can be lowered (for 
example, by adding a small amount of nitric oxide or NO) but at the penalty of 
a higher vapor pressure and slightly reduced performance. This mixture of NO 
and N2O4 is called mixed oxides of nitrogen (MON) and different grades have 
between 2 and 30% by weight NO content. 

Nitrogen tetroxide oxidizer is used with UDMH in many Russian engines and 
almost all their small thrusters. It is also used with monomethylhydrazine fuel in 
the Space Shuttle orbital maneuver system, multithruster reaction control system, 
and in many U.S. spacecraft propulsion systems. In many of these applications 
care must be taken to avoid freezing this nitrogen tetroxide. The OSHA 8-hr 
personnel exposure limit is 5 ppm or 9 mg/vp? . 



7.3. LIQUID FUELS 

Again, many different chemicals have been proposed, investigated, and tested 
as fuels. Only a few have been used in production rocket engines. Liquid fuels 
other than those listed in this chapter have been used in experimental rocket 
engines, in older experimental designs, and in some older production engines. 
These include aniline, furfuryl alcohol, xylidine, gasoline, hydrazine hydrate, 
borohydrides, -methyl and/or ethyl alcohol, ammonia, and mixtures of some of 
these. 

Hydrocarbon Fuels 

Petroleum derivatives encompass a large variety of different hydrocarbon chem- 
icals, most of which can be used as a rocket fuel. Most common are those types 
that are in use with other applications and engines, such as gasoline, kerosene, 
diesel oil, and turbojet fuel. Their physical properties and chemical composi- 
tion vary widely with the type of crude oil from which they were refined, with 
the chemical process used in their production, and with the accuracy of control 
exercised in their manufacture. Typical values are listed in Table 7-2. 

In general, these petroleum fuels form yellow-white, brilliantly radiating 
flames and give good performance. They are relatively easy to handle, and there 



260 LIQUID PROPELLANTS 



TABLE 7-2. Properties of Some Typical Hydrocarbon Fuels Made from Petroleum 





Jet 

Fuel 


Kerosene 


Aviation 

Gasoline 

100/130 


RP-1 


RP-2 


Specific gravity at 


0.78 


0.81 


0.73 


0.80-0.815 


0.80-0.815 


289 K 


Freezing point (K) 


213 (max.) 


230 


213 


213 (max.) 


213 (max) 


Viscosity at 289 K 


1.4 


1.6 


0.5 


16.5 (at 239 K) 


16.5 (at 239 K) 


(cP) 


Flash point" (K) 


269 


331 


244 


333 


333 


ASTM distillation 
(K) 


10% evaporated 


347 


— 


337 


458-483 


458-483 


50% evaporated 


444 


— 


363 


— 


— 


90% evaporated 


511 


— 


391 


— 


— 


Reid vapor 


2 to 3 


Below 1 


7 


— 


— 


pressure (psia) 


Specific heat 


0.50 


0.49 


0.53 


0.50 


0.50 


(cal/kg-K) 


Average molecular 


130 


175 


90 


— 


— 


mass (g/mol) 


Sulfur, total mg/kg 


— 


— 


— 


30 (max) 


1 (max) 



"Tag closed cup method. 



is an ample supply of these fuels available at relatively low cost. A specifically 
refined petroleum product particularly suitable as a rocket propellant has been 
designated RP-1 (Rocket Propellant number 1). It is basically a kerosene-type 
fuel mixture of hydrocarbons with a somewhat narrow range of densities and 
vapor pressure. Several hydrocarbon fuels can form carbon deposits on the inside 
of cooling passages, impeding the heat transfer and raising wall temperatures. 
Ref. 7-7 indicates that this carbon formation depends on fuel temperature in 
the cooling jacket, the particular fuel, the heat transfer, and the chamber wall 
material. RP-1 is low in olefins and aromatics, which can cause carbonaceous 
deposits inside fuel cooling passages. RP-1 has been used with liquid oxygen 
in many early rocket engines (see Figs. 5-1 to 5-6 and Ref. 7-8). A very 
similar kerosene-type fuel is used in Russia today. A few years ago RP-2 was 
substituted for RP-1 in U.S. rocket applications. The principal difference is a 
reduced sulphur content (see Table 7-2), because this impurity was believed to 
have caused corrosion in cooling jackets of certain thrust chambers. See Ref. 
7-8. 

Methane (CH 4 ) is a cryogenic hydrocarbon fuel. It is denser than liquid hydro- 
gen and relatively low in cost. Compared to petroleum-refined hydrocarbons, it 
has highly reproducible properties. With liquid oxygen it is a candidate pro- 
pellant combination for launch vehicle booster rocket engines and also reaction 



7.3. LIQUID FUELS 261 



control engines when oxygen is available from the main engines. Experimental 
oxygen -methane engines have been tested, but they have not yet flown. 



Liquid Hydrogen (H2) 

Liquid hydrogen, when burned with liquid fluorine or liquid oxygen, gives a high 
performance, as shown in Table 5-5. It also is an excellent regenerative coolant. 
With oxygen it burns with a colorless flame; however, the shock waves in the 
plume may be visible. Of all known fuels, liquid hydrogen is the lightest and 
the coldest, having a specific gravity of 0.07 and a boiling point of about 20 K. 
The very low fuel density requires bulky fuel tanks, which necessitate very large 
vehicle volumes. The extremely low temperature limits the selection of materials 
for pumps, cooling jackets, tanks, and piping because many metals become brittle 
at low temperatures. 

Because of its low temperature, liquid hydrogen tanks and lines have to be 
well insulated to minimize the evaporation of hydrogen or the condensation of 
moisture or air on the outside with the subsequent formation of liquid or solid air 
or ice. A vacuum jacket often has been used in addition to insulating materials. 
All common liquids and gases solidify in liquid hydrogen. These solid particles in 
turn plug orifices and valves. Therefore, care must be taken to purge all lines and 
tanks of air and moisture (flush with helium or pull vacuum) before introducing 
the propellant. Mixtures of liquid hydrogen and solid oxygen or solid air can be 
explosive. 

Liquid hydrogen has two isomers, namely, orthohydrogen and parahydrogen, 
which differ in the orientation of their nuclear spin state. As hydrogen is liquefied 
and cooled to lower and lower temperatures, the relative equilibrium composition 
of ortho- and parahydrogen changes. The transformation from orthohydrogen to 
parahydrogen is exothermic and results in excessive boiloff, unless complete 
conversion to parahydrogen is achieved during liquefaction. Liquid hydrogen is 
manufactured from gaseous hydrogen by successive compression, cooling, and 
expansion processes. 

Hydrogen gas, when mixed with air, is highly flammable and explosive over 
a wide range of mixture ratios. To avoid this danger, escaping excess hydrogen 
gas (at tank vent lines) is often intentionally ignited and burned or “flared” in 
air. Liquid hydrogen is used with liquid oxygen in the Delta IV launcher. Cen- 
taur upper stage, the Space Shuttle main engine, and upper stage space engines 
developed in Japan, Russia, Europe, India, and China. 

Hydrogen burning with oxygen forms a nontoxic almost invisible exhaust 
gas. This propellant combination has been applied successfully to space launch 
vehicles because of its high specific impulse. Here the payload capability usually 
increases greatly for relatively small increases in specific impulse. However, the 
low density of hydrogen makes for a very large fuel tank, a large vehicle, and a 
relatively high drag. 

Some studies have shown that, when burned with liquid oxygen, a hydrocarbon 
(such as methane or RP-1) can give a small advantage in space launch vehicle first 



262 LIQUID PROPELLANTS 



stages. Here the higher average propellant density allows a smaller vehicle with 
lower drag, which compensates for the lower specific impulse of the hydrocarbon 
when compared to a hydrogen fuel. Also, there are some concepts for operating 
the booster-stage rocket engine initially with hydrocarbon fuel and then switching 
during flight to hydrogen fuel. Engines using LOX and these two fuels, namely 
hydrocarbon and hydrogen, have been called tripropellant engines. They have 
not yet been fully developed or flown. Some work on experimental engines was 
done in Russia, but there is no known current effort. 



Hydrazine (N2H4) 

Reference 7-9 gives a good discussion of this propellant, which is used as a 
bipropellant fuel as well as a monopropellant. Hydrazine, monomethylhydrazine 
(MMH), and unsymmetrical dimethylhydrazine (UDMH) have similar physical 
and thermochemical properties. Hydrazine is a toxic, colorless liquid with a high 
freezing point (274.3 K or 34°F). Hydrazine tanks, pipes, injectors, catalysts, and 
valves are usually electrically heated to prevent freezing in cool weather or in 
outer space. Hydrazine has a short ignition delay and is spontaneously ignitable 
with nitric acid, nitrogen tetroxide, and concentrated hydrogen peroxide. 

Its vapors may form explosive mixtures with air. If hydrazine is spilled on a 
porous surface or a cloth, a spontaneous ignition with air can occur. 

Pure anhydrous hydrazine is a stable liquid; it has been safely heated above 480 
K. It has been stored in sealed tanks for over 15 years. With impurities or at higher 
temperatures it decomposes and releases energy. Under pressure shock (blast 
wave or adiabatic compression) hydrazine vapor or hydrazine mist can decom- 
pose at temperatures as low as 367 K. Under some conditions this decomposition 
can be a violent detonation, and this has caused problems in cooling passages 
of experimental injectors and thrust chambers. Harmful effects to personnel may 
result from ingestion, inhalation of vapors, or prolonged contact with skin. The 
American Conference of Government Industrial Hygienists (ACGIH) recom- 
mended 8-hr personnel exposure limit is 0.01 ppm or 0.013 mg/m^. Hydrazine 
is a known animal carcinogen and a suspected carcinogen for people. 

Hydrazine reacts with many materials, and care must be exercised to avoid 
storage contact with materials that cause a decomposition (see Refs. 7-9 and 
7-10). Tanks, pipes, injectors, catalysts, or valves must be cleaned and free of 
impurities. Compatible materials include certain stainless steels (303, 304, 321, 
or 347), nickel, and 1100 and 3003 series of aluminum. Iron, copper, and its 
alloys (such as brass or bronze), monel, magnesium, zinc, and some types of 
aluminum alloy must be avoided. 



Unsymmetrical Dimethylhydrazine [(CHs)2NNH2] 

A derivative of hydrazine, namely UDMH, is often used instead of or in mix- 
tures with hydrazine because it forms a more stable liquid, particularly at higher 
temperatures. Furthermore, it has a lower freezing point (215.9 K) and a lower 



7.4. LIQUID MONOPROPELLANTS 263 



boiling point (336.5 K) than hydrazine. When UDMH is burned with an oxidizer 
it gives only slightly lower values of 4 than pure hydrazine. UDMH is often used 
when mixed with 30 to 50% hydrazine or with 25% hydrazine hydrate. UDMH 
is used in Russian and Chinese rocket engines and small thrusters. The ACGIH 
recommended 8 -hr personnel exposure limit for vapor is 0.01 ppm, and UDMH 
is a suspected carcinogen. 



Monomethylhydrazine (CH 3 NHNH 2 ) 

This fuel abbreviated as MMH has been used extensively in U.S. spacecraft 
rocket engines, particularly in small attitude control engines, usually with N 2 O 4 
as the oxidizer. It has a better shock resistance to blast waves and a better 
liquid temperature range than pure hydrazine. Like hydrazine, its vapors are eas- 
ily ignited in air; the flammability limits are from 2.5 to 98% by volume at 
atmospheric sea-level pressure and ambient temperature. The materials compat- 
ible with hydrazine are also compatible with MMH. The specific impulse with 
storable oxidizers usually is 1 or 2% lower with MMH than with N 2 H 4 . 

Both MMH and UDMH are soluble in many hydrocarbons; hydrazine is not. 
All hydrazines are toxic materials, but MMH is the most toxic when inhaled. 
Atmospheric concentrations of all hydrazines should be kept below 0.1 ppm for 
long periods of personnel exposure. 

Monomethylhydrazine, when added in relatively small quantities of 3 to 15% 
to hydrazine, has a substantial quenching effect on the explosive decomposition of 
hydrazine. Monomethylhydrazine decomposes at 491 K, whereas hydrazine can 
explode at 369 K when subjected to certain pressure shocks. MMH is a known 
animal carcinogen and the ACGIH recommended personnel 8 -hour exposure limit 
is 0.01 ppm. 



7.4. LIQUID MONOPROPELLANTS 

The propellant-feed and control-system simplicity associated with a monopropel- 
lant makes this type of propellant attractive for certain applications. Hydrazine 
is being used extensively as a monopropellant in small attitude and trajectory 
control rockets for the control of satellites and other spacecraft and also as a 
hot gas generator. (It is discussed in the preceding section.) Other monopro- 
pellants (ethylene oxide or nitromethane) were tried experimentally, but are no 
longer used today. Concentrated hydrogen peroxide was used for small thrusters 
between 1945 and 1965 and for monopropellant gas generation in the United 
States, Russia, United Kingdom or Britain, and Germany. 

Ignition of monopropellants can be achieved thermally (electrical or flame 
heat) or by a catalytic material. A monopropellant must be chemically and ther- 
mally stable to ensure good liquid storage properties, and yet it must be easily 
decomposed and reactive to quickly give good combustion. 



264 LIQUID PROPELLANTS 



Hydrazine as a Monopropellant 

Hydrazine is not only an excellent storable fuel but also an excellent monopro- 
pellant when decomposed by a suitable solid catalyst; this catalyst often needs 
to be preheated for fast startup or for extending the useful catalyst life. Iridium 
on porous alumina base is an effective catalyst at room temperature. At ele- 
vated temperature (about 450 K) many materials decompose hydrazine, including 
iron, nickel, and cobalt. See Ref. 7-9. Different catalysts and different catalyst 
configurations make the decomposition reaction go to different products, result- 
ing in gases varying in composition or temperature. As a monopropellant, it 
is used in gas generators or in space engine attitude control rockets. A typical 





e 

o 

o 



80 
60 
40 
20 

0 20 40 60 80 100 

Ammonia decomposition, % 




FIGURE 7-3. Operating parameters for decomposed hydrazine at the exit of a catalytic 
reactor as a function of the ammonia dissociation fraction. Adapted with permission from 
Ref. 7-9. 



7.5. GELLED PROPELLANTS 265 



hydrazine monopropellant thrust chamber, its injection patterns, and its decom- 
position reaction are described in Fig. 8-14. Typical operating parameters are 
shown in Fig. 7-3. 

The catalytic decomposition of hydrazine can be described ideally as a two- 
step process; this ignores other steps and intermediate products. First, hydrazine 
(N2FI4) decomposes into gaseous ammonia (NFI3) and nitrogen (N2); this reac- 
tion is highly exothermic, that is, it releases heat. Second, the ammonia decom- 
poses further into nitrogen and hydrogen gases, but this reaction is endothermic 
and absorbs heat. These simplified reactions can be written as 

3N2H4 ^ 4(1 - x)NH 3 -h (1 -h 2 x)N 2 -h 6.rH2 (7-4) 

Flere x is the degree of ammonia dissociation; it is a function of the catalyst type, 
size, and geometry, the chamber pressure, and the dwell time within the catalyst 
bed. Figure 7-3 shows several ideal rocket engine parameters for hydrazine 
monopropellant as a function of x, the fraction of ammonia that is decomposed. 
The values are for an ideal thruster at 1000 psia chamber pressure with a nozzle 
area ratio of 50 expanding at high altitude. The best specific impulse is attained 
when little ammonia is allowed to dissociate. 



7.5. GELLED PROPELLANTS 

Gelled propellants have additives that make them thixotropic materials. They 
have the consistency of thick paint or jelly when at rest, but they liquefy and 
flow through pipes, valves, pumps, or injectors when an adequate shear stress is 
applied. They offer these advantages: (1) Small aluminum fuel particles can be 
suspended in the fuels. Inert solid particles can be suspended in oxidizer liquids. 
This increases propellant density, density impulse, and thus reduces the size of 
tanks and vehicles. (2) There is no plugging of injector orifices or valve passages 
and good flow control has been demonstrated. (3) Most gelled fuel propellants are 
essentially nonflammable and will not usually sustain an open fire. (4) There is 
reduced susceptibility of leakage or spill, reduced sloshing of liquids in the tanks, 
and the boil-off rate is reduced. (5) Long-term storage of 10 years or more without 
settling or separation of the solid particles has been demonstrated. (6) Explosions 
or detonations, which happen when a vehicle accident causes liquid propellants to 
become inadvertently mixed, are much less likely with gelled propellants, which 
are difficult to mix. (7) Short-duration pulsing has been tested. (8) Most storable 
oxidizers, a few cryogenic propellants, and most liquid storable fuels can be 
gelled. (9) Explosions are much less likely when a propellant tank is penetrated 
by a bullet or when a missile is exposed to an external fire or a nearby detonation. 

These are some of the disadvantages: (1) There is a small decrease in specific 
impulse due to dilution with a gelling agent, and less efficient atomization into 
small droplets (due to high viscosity) or less efficient combustion. Eor example, 
the characteristic velocity c* of oxygen-kerosene propellant is decreased by 



266 LIQUID PROPELLANTS 



4 to 6% when the kerosene is gelled and aluminum is suspended in the fuel. 

(2) Bubhle-free loading or unloading of propellants is somewhat more complex 
and may take longer. Residual propellant quantity will be slightly higher because 
the thixotropic fluid layer on the walls of the tanks and pipes will be thicker. 

(3) Changes in ambient temperature will cause slight changes in propellant density 
and viscosity and therefore also in mixture ratio; this can result in more leftover 
or residual propellant and thus in a slight reduction of available total impulse. 
Suspended metals can make the exhaust plume smoky and visible. 

Experimental rocket engines have shown these gelled propellants to be gen- 
erally safer than ordinary liquid propellants and to have adequate performance 
and operational characteristics (see Refs. 7-11 and 7-12). This makes them less 
susceptible to field accidents. A variety of different organic and inorganic gelling 
agents have been explored with a number of different liquid propellants. 

Experimental thrust chambers and rocket engine systems have been satisfacto- 
rily demonstrated with several gelled propellant combinations. One experimental 
engine has flown. As far as is known, no such rocket engine has yet been put 
into production or flight operation. 



7.6. GASEOUS PROPELLANTS 

Cold gas propellants have been used successfully for reaction control systems 
(RCS) for more than 60 years. The words “cold gas” (stored at ambient temper- 
ature) is distinguished from a “warm gas,” which has been heated. The engine 
system is simple, consisting of one or more high-pressure gas tanks, multiple 
simple nozzles (often aluminum or plastic), an electrical control valve with each 
nozzle, a pressure regulator, and provisions for filling and venting the gas. The 
tank size will be smaller if the tank pressures are high. Pressures are typically 
between 300 and 1000 MPa (about 300 to 10,000 psi), and these gas tanks have 
thick wells and are heavy. 

Typical cold gas propellants and some of their properties and characteristics 
are listed in Table 7-3. Nitrogen, argon, dry air, krypton and Freon 14 have 
been employed in spacecraft RCS. With high-pressure hydrogen or helium as 
cold gas, the specific impulse is much higher, but the densities of these gases are 
much lower. This requires a much larger gas storage volume and heavier high- 
pressure tanks. In most applications the extra inert mass outweighs the advantage 
of better performance. In a few applications the gas (and its storage tank) are 
heated electrically or chemically. This improves the specific impulse and allows 
a smaller tank, but it also introduces complexity. 

The selection of the gas propellant, the storage tanks, and RCS design depend 
on many factors, such as volume and mass of the storage tanks, the maximum 
thrust and total impulse, the gas density, required maneuvers, duty cycle, and 
flight duration. Cold gas systems have been used for total impulses of perhaps 
22,200 N-sec or 5000 Ibf-sec. Higher values usually employ liquid propellants. 

If the operation is short (most of the gas consumed in a few minutes, while 
the main engine is running), the gas expansion will be close to adiabatic (no heat 



7.7. SAFETY AND ENVIRONMENTAL CONCERNS 267 



TABLE 7-3. Properties of Gaseous Propellants Used for Auxiliary Propulsion 



Propellant 


Molecular 

Mass 


Density" 

(lb/ft3) 


Specific heat 
ratio k 


Theoretical Specific 
Impulse* (sec) 


Hydrogen 


2.0 


1.77 


1.40 


284 


Helium 


4.0 


3.54 


1.67 


179 


Methane 


16.0 


14.1 


1.30 


114 


Nitrogen 


28.0 


24.7 


1.40 


76 


Air 


28.9 


25.5 


1.40 


74 


Argon 


39.9 


35.3 


1.67 


57 


Krypton 


83.8 


74.1 


1.63 


50 



“At 5000 psia and 20“C. 

*In vacuum with nozzle area ratio of 50:1 and initial temperature of 20°C. 



absorption by gas) and often is analyzed as isentropic. The temperature of the 
gas, the pressure, and specific impulse will drop as the gas is consumed. For long 
intermittent operations (months or years in space) the heat from the spacecraft 
is transfered to the gas and the tank temperature stays essentially constant; the 
expansion will be nearly isothermal. 

The advantages and disadvantages of cold gas thrusters and systems are 
described in Section 8.3 in the discussion of low thrust. 



7.7. SAFETY AND ENVIRONMENTAL CONCERNS 

To minimize the hazards and potential damage inherent in reactive propellant 
materials, it is necessary to be very conscientious about the likely risks and 
hazards (see Refs. 7-4, 7-13, and 7-14). This concerns toxicity, explosiveness, 
fire or spill danger, and others mentioned in Section 7.1. Before an operator, 
assembler, maintenance mechanic, supervisor, or engineer is allowed to transfer 
or use a particular propellant, he or she should receive safety training in the 
particular propellant, its characteristics, its safe handling or transfer, potential 
damage to equipment or the environment, and the countermeasures for limiting 
the consequences in case of an accident. They must also understand the potential 
hazards to the health of personnel, first aid, remedies in case of contact exposure 
of the skin, ingestion, or inhaling, and the use of safety equipment. Examples of 
safety equipment are protective clothing, face shields, detectors for toxic vapors, 
remote controls, warning signals, or emergency water deluge. The personnel 
working with or being close to highly toxic materials usually have to undergo 
periodic health monitoring. Also rocket engines need to be designed for safety 
to minimize the occurrence of a leak, an accidental spill, an unexpected fire, or 
other potentially unsafe conditions. Most organizations have one or more safety 
specialists who review the safety of the test plans, manufacturing operations, 
designs, procedures, or safety equipment. With the proper training, equipment, 
precautions, and design safety features, all propellants can be handled safely. 



268 LIQUID PROPELLANTS 



If a safety violation occurs or if an operation, design, procedure, or practice 
is found to be (or appears to be) unsafe, then a thorough investigation of the 
particular item or issue should be undertaken, the cause of the lack of safety 
should be investigated and identified, and an appropriate remedial action should 
be selected and initiated as soon as possible. 

The discharge of toxic exhaust gases to the environment and their dispersion 
by the wind can cause exposure of operating personnel as well as the people in 
nearby areas, and can cause damage to plants and animals. This is discussed in 
Section 20.2. The dumping or spilling of toxic liquids can contaminate subter- 
ranean aquifers and surface waters, and their vapors can pollute the air. Today 
the type and amount of gaseous and liquid discharges are regulated and moni- 
tored by government authorities. These discharge quantities must be controlled 
or penalties will be assessed against violators. Obtaining a permit to discharge 
can be a lengthy and involved procedure. 



PROBLEMS 

1 . Plot the variation of the density specific impulse (product of average specific gravity 
and specific impufse) with mixture ratio and expfain the meaning of the curve. Use the 
theoretical shifting specific impulse values of Fig. 5-1 and the specific gravities from 
Fig. 7-1 or Table 7-1 for the liquid oxygen-RP-1 propellant combination. 

Answers: Check point at r = 2.0; = 290; Id = 303; i5av = 1-01. 

2 . Prepare a table comparing the relative merits of liquid oxygen and nitrogen tetroxide 
as rocket engine oxidizers. 

3 . Derive Eq. 7-1 for the average specific gravity. 

4 . A rocket engine uses liquid oxygen and RP-1 as propellants at a design mass mixture 
ratio of 2.40. The pumps used in the feed system are basically constant-volume flow 
devices. The RP-1 hydrocarbon fuel has a nominal temperature of 298 K and it can 
vary at about ±25°C. The liquid oxygen is nominally at its boiling point (90 K), 
but, after the tank is pressurized, this temperature can increase by up to 30 K during 
long time storage. What are the extreme mixture ratios under unfavorable temperature 
conditions? If this engine has a nominal mass flow rate of 100 kg/sec and a duration 
of 100 sec, what is the maximum residual propellant mass when the other propellant 
is fully consumed? Use the curve slopes of Fig. 7-1 to estimate changes in density. 
Assume that the specific impulse is constant for the relatively small changes in mixture 
ratio, that small vapor pressure changes have no influence on the pump flow, that there 
is no evaporation of the oxygen in the tank, and that the engine has no automatic 
control for mixture ratio. Assume zero residual propellant. 

5 . The vehicle stage propelled by the rocket engine in Problem 4 has a design mass ratio 
nif/nio of 0.50 (see Eq. 4-6). For the specific impulse please use a value half way 
between the shifting and the frozen equilibrium curves of Fig. 5-1. How much will 
the worst combined changes in propellant temperatures affect the mass ratio and the 
ideal gravity-free vacuum velocity? 



REFERENCES 269 



6. (a) What should he the approximate percent ullage volume for nitrogen tetroxide tank 

when the vehicle is exposed to ambient temperatures between about 50°F and 
about 150°F? 

(b) What is maximum tank pressure at 150°F. 

(c) What factors should be considered in part (b)? 

Answers: (a) 15 to 17%; the variation is due to the nonuniform temperature distribution 
in the tank; (b) 6 to 7 atm; (c) vapor pressure, nitrogen monoxide content in the 
oxidizer, chemical reactions with wall materials, or impurities that result in largely 
insoluble gas products. 

7. An insulated, long vertical, vented liquid oxygen tank has been sitting on the sea-level 
launch stand for a period of time. Assume no heat loss to the tank wells. The surface 
of the liquid is at atmospheric pressure and is 10.2 m above the closed outlet at the 
bottom of the tank. If there is no circulation, what will be the temperature, pressure, 
and density of the oxygen at the tank outlet? 



SYMBOLS 

III density specific impulse, sec 

7j specific impulse, sec 

k ratio of specific heats 

r mixture ratio (mass flow rate of oxidizer to mass flow rate of fuel) 



Greek Letters 



^av 

Pav ■: Pf ■: Po 



average specific gravity of mixture 
specific gravity of fuel 
specific gravity of oxidizer 

densities average, fuel, and oxidizer, kg/m^ (Ibm/ft^) 



REFERENCES 

7-1. S. F. Samer, Propellant Chemistry, Reinhold Publishing, New York, 1966. 

7-2. Chemical Rocket Propellant Hazards, Vol. 1, General Safety Engineering Design 
Criteria, Chemical Propulsion Information Agency (CPI A) Publication 194, Octo- 
ber 1971. 

7-3. L. C. Sutherland, “Scaling Law for Estimating Liquid Propellant Explosive Yields,” 
Journal of Spacecraft and Rockets, March-April 1978, pp. 124-125. 

7-4. NIOSH Pocket Guide to Chemical Hazards, DHHS (NIOSH) Publication No. 2005- 
149, Department of Health and Human Services, Washington DC, September 2005. 

7-5. 1990-1991 Threshold Limit Values for Chemical Substances and Physical Agents 
and Biological Exposure Indices, American Conference of Government Industrial 
Hygienists, Cincinnati, OH, 1990 (revised periodically). 



270 



LIQUID PROPELLANTS 



7-6. H. E. Malone, The Analysis of Rocket Propellants , Academic Press, New York, 
1976. 

7-7. K. Liang, B. Yang, and Z. Zhang, “Investigation of Heat Transfer and Coking 
Characteristics of Hydrocarbon Fuels,” Journal of Propulsion and Power, Vol. 14, 
No. 5, September-October 1998. 

7-8. Detail Specification — Propellant Rocket Grade Kerosene, Department of Defense, 
Washington DC, M1L-DTL-25576D, 20 May 2005. 

7-9. E. W. Schmidt, Hydrazine and Its Derivatives, Preparation, Properties, Applica- 
tions, 2nd ed., John Wiley & Sons, New York, 2001. 

7-10. O. M. Morgan and D. S. Meinhardt, “Monopropellant Selection Criteria — 
Hydrazine and other Options,” AIAA Technical Paper 99-2595, June 1999. 

7-11. K. F. Hodge, T. A. Crofoot, and S. Nelson, “Gelled Technical Propellants for 
Tactical Missile Application,” AIAA Technical Paper 99-2976, June 1999. 

7-12. B. D. Allen, “History, Development and Testing of Thixotropic Gels for Advanced 
Systems,” AIAA Propulsion Conference, July 1985. 

7-13. Hazards of Chemical Rockets and Propellants , Vol. I, Safety, Health and the 
Environment , and Vol. Ill, Liquid Propellants , Chemical Propulsion Information 
Analysis Center, CPIA Publication 394, 1984. 

7-14. Emergency Response Guidebook, National Institute for Occupational Safety and 
Health (NIOSH), Department of Health and Human Services, Washington, DC, 
rev. ed., 2004. 



CHAPTER 8 



THRUST CHAMBERS 



This chapter describes thrust chambers, their components, cooling, ignition, and 
heat transfer. The thrust chamber is the key subassembly of a rocket engine. 
Here the liquid propellants are metered, injected, atomized, vaporized, mixed, 
and burned to form hot reaction gas products, which in turn are accelerated and 
ejected at supersonic velocity (see Refs. 6-1, 6-2, and 8-1). A rocket thrust 
chamber assembly (Figs. 8-1 and 8-2) has an injector, a combustion chamber, 
a supersonic nozzle, and mounting provisions. All have to withstand the extreme 
heat of combustion and the various forces, including the transmission of the 
thrust force to the vehicle. There also is an ignition system if nonspontaneously 
ignitable propellants are used. Some thrust chamber assemblies also have inte- 
grally mounted propellant valves and sometimes a thrust vector control device, 
as described in Chapter 18. Table 8-1 gives various data about five different 
thrust chambers with different kinds of propellants, cooling methods, injectors, 
feed systems, thrust levels, or nozzle expansions. Some engine parameters are 
also listed. Some of the terms used in this table will be explained later in this 
chapter. 

The basic analyses for thrust chamber specific impulse and combustion temper- 
ature are given in Chapter 5, the basic design parameters (thrust, flow, chamber 
pressure, or throat area) are in Chapter 3, and the combustion phenomena in 
Chapter 9. 

Although we use the word thrust chamber in this book (for rocket engines 
generally larger than 1000 Ibf thrust), some technical publications use the terms 
thrust cylinder , thrust cell, or rocket combustor . We will also use the term thruster 
for small thrust units, such as attitude control thrusters, and for electrical propul- 
sion systems. 



271 



272 THRUST CHAMBERS 




FIGURE 8-1. Construction of an early regeneratively cooled tubular thrust chamber 
using a kerosene-type fuel and liquid oxygen, as originally used in the Thor missile. 
The nozzle throat inside diameter is about 15 in. The sea-level thrust was originally 
120,000 Ibf, hut was uprated to 135,000, then 150,000, and finally to 165,000 Ibf by 
increasing the flow and chamber pressure and strengthening and modifying the hardware. 
The cone-shaped nozzle exit was replaced by a bell-shaped nozzle exit. Figure 8-9 shows 
how the fuel flows down through every other tube and returns through the adjacent tube 
before flowing into the injector. Figure 8-5 shows the flow passages in a similar injector. 
(Courtesy of Rolls Royce, England.) 



r — Stainless steel inner 
\ liner with 120 groove 
1 2 quarter-wave \ coolant passages and 




273 



MGUKli 8-2. Simplmed halt-section ot one ot the two thrust chambers ot the orbital maneuvering engines used on the Space Shuttle Orbiter. 
Each develops a vacuum thrust of 6000 Ibf (26,689 N) and delivers a minimum vacuum specific impulse of 310 sec, using nitrogen tetroxide 
and monomethyl hydrazine propellants at a nominal mixture ratio of 1.65 and a nominal chamber pressure of 128 psia. It is designed for 
100 flight missions, a service life of 10 years, and a minimum of 500 starts. These engines provide the thrust for final orbit attainment, orbit 
circularization, orbit transfer, rendezvous, and deorbit maneuvers. The nozzle area ratio is 55:1. (Courtesy of Aerojet Propulsion Company.) 



TABLE 8-1. Thrust Chamber Characteristics 



T3 

o c 

CJ ^ - 



O -Si, C - 

O o 

^ U lyj 



C 

^ II 5 

c /3 



Q 



“ifis 

-Ills 

H C bO 

1-^ . . O 

iS ^ ^ 3 o 

u 3 ti 8 c 

Q j 2 o -S 



c 

.2 

c 

c/3 

(U 

Q 



bO 

C 

tP 



-o o 
D h 
•S C3 
s o 



d 

cb 


X 




2 

8 

2.0 


o 


o 


o 


Dh 




O 


1— ^ 


o 


c^ 


00 


S 

o 

U 


n 

z 




cs 


(N 


in 



u ^ 
Mo g 

s ^ K 



CJ 






w 



D 

x; c/3 
c/3 p 

« IS 
2 K 3 3 

o hM c/3 

CQ u 
.P ^ 



c 

o 

T3 '5 

> >. B S 

M 3-3 0 

" O C 3 pi. 

■d M .-3 ^ 3 

S 3-3.3 

3 > d U 

K tH ^ F “ 
n o o>) d 

ii g-^ 3 a 

3 3-300 

« = 2 ^ o 
D CL, o c 
QC -3 



Dh 



a 

U 






o 

IT) 

X 

(N 

o 



o 

0\ 



O 

(N 

z 



^ o 
<C >o 
Z 



m 

'sO 

O o ^ 
p) g ^ ^ 
^ VO 



r- 

m 



in 
































(U 


c 






in 








c 


CJ 






cn 








(U 

c/3 


bO 


o 


o 


CN 


r- 




O 


o 


X 


o 


o 




»n 


ON 




u, 

(U 


O 

"D 


i> 


r-" 




CN 


(N 


in 

in 








o 












'3 




CN 











& 



cr 

3 



m 



O 00 
r-. 'so 



X 


o 


o 

o 


o 

o 








in 


'3 


’3 


o' 


CN 


O' 


cr 


ON 




X 


J 




CN 



ON 


NO 


00 


ON 


ON 


in 




NO 






ON 




cn 




in 


cn 






in 





o^ § 



bO 

C 



(N 


(N 






00 








CN 


X 


o 




o 


in 








;o 




o 

> 


in 


c 


NO 

NO 


00 

l> 


cn 

»n 


'3 

O' 


’3 

cr 


0 

1 

cb 


(N 




z 




in 

r- 




3 


3 


D 

c/3 















^ < 
^ Z 



o 

»o 



o 

Z 



^ g'ct S 



.2 g. 

2 M 

D O 

3 S 
>< 



_ c? ct 
.2-5 I 

O 2 M 

b 3 s 

D Cfl > 
c/3 " 



D ^ 
Dh CJ 

c/3 1) 



D 

CJ -C 

(U X 

c/3 1) 







X) 

3 


o 

> 


3 


■g 


X 

3 


(J 

> 


S 






cb 


D 


D 


cb 


3 


(J 


3 






X 




3 


X 


X 




3 




1-H 

CJ 


CJ 

c/3 


Cb 

D 

c/3 


CJ 

cb 

> 


CJ 

c/3 


CJ 

c/3 


3 

CJ 

c/3 


CJ 

3 

> 






3 


1 . 




3 


3 


) 1 




o 

D 


'x 




Cb 






u 

X 


3 





■■^ 's 

c/3 ^ 



^ ^ 
(U u — ! 

ti Q. ^ 

Oh > 

i- C 

3 -2 A 
■§3 3 
I c “ 

3 6J0 tH 

x: (U 

3 t2 -3 

■2 -o i 



cb c 






:b c/3 

s 

=9 

U H 



274 



Gimbal mount sea-level dry weight (Ibf) < 10 57.3 NA 70 23 

Chamber diameter (in.) — 15.75 1.09 21 11.7 

Nozzle throat diameter (in.) 5.2 9.25 0.427 16.2 7.5 

Nozzle exit diameter (in.) 88 68.28 3.018 45.8 60 

Nozzle exit area ratio 285 54:1 50:1 8:1 65:1 



IT) r-; 
O 00 

CO — 



D O 



»o 



cd 



in 

3 

< 



o 

c 

CL> 

cd 

CJ 



< 



I 

o ^ 



’Qh ^ 



*0 ^ 
2 •§ 
j= g 
•a 



o o o o S 

g -t ^ g g 

T3 ^ 

U 

c 

3 



s 

V5 cd 



Oh ^ 



00 ^ ^ 
'sO CO 00 



o 

Z 



00 

.s 

’o 

o 

o uo 
> o 



CJ 

J=> 

D 



't-i cd 
T3 



CnJ o 
o^ o 

CN ^ 



-o 

c 

cd 

V5 

00 

c 



tin 



O »/l o 

^ CO 
'O — in 
in ^ 



Q- S 

E S 

s ^ 

^ 00 

H 



^ 00 



D 

c 

o 

z 



i> 00 in 
O ■^* CO 



i 

^ .S 
o 

o 2 
o ‘P 



o 

Z 



CJ 

a 

CJ 

00 

(U 

Qi 



V5 

o < = 

Z A ^ 

Q 



T3 

C 

cd 



00 o 
00 

cN in 



^ 't; 



o ' 

Oh 



'o 

K 



o o 
»n in 



r- 



CS (N 
»n in 



== s 

V3 Cd 
V5 ■*-' 

^ X3 

cu ^ 



Q- 

E 

D 

Q- 

O 

X5 

u 

e 2 



o 

o^ 



c 

o 

X) 

u 

cd 



OJ 

X 

D 



-- O 

^ X 
I CO X 

3 A ^ 



o 

X 

u 

cd 

u 






JJ 



:§ ^ < 
r o 

c 

CJ 

00 

(U 



< »n 

Z 



-fa 

g s 

o > 
U S 



2 ^ X X 
^ in ^ 
CS (N 



> CJ 
^ >> 
Q- «-• 

P Vh 
D- r- 

o 2 

X Q, 
D & 

e 2 “ 



o 

•s J3 
g M 
3 g 

(U 35 

^ g 
C -S 
° S 

Cd 



a 



CJ 



c 

o 



cd cd 
X X 

U U 



00 o 
c > 

1 — I 73 

"o 

cd 

^ 's 

o a 

g ■§ 
•-§ § 
! S •<-' 
. Cd zJ 

1 CJ 

t .;_, cd 

S "a 

H Ph 



D 
"O 
00 
C .' 



c« X 

c ^ 



CJ 



N 



D 

^ - 
z u 



X ^ 

5 « 

3^ ^ 

^ 00 
■p c 

cd X 
tc O 

(3 O 

U 



X 

X 

'5 



c 

c 

cd 

X 



E 

cd 

X 

CJ 

X 

e 2 



& 

o 



X ^ 



! 3 o 

, “ ^ 

) o ?o 

S-( 

; fa- i-i 

^ B 

: a; 

• cd c 



O o 



ft o-.S’ 

2 S i- 
-d -3 “ 
D X 

fa 3 



(U CJ 

V- l- 

== =3 ^ 

M IZ2 

M C/3 Cj 

a a ‘S 

ft ft ° 

^ fa 

2 o “ 



'E 

3 

z 



u 



00 

s 

PJ 



Uh 



Engine thrust (at sea level) (lb) NA 190,400 12 165,000 NA 

Engine thrust (altitude) (lb) 24,750 242,500 18 207,700 9850 

Engine specific impulse at sea level NA 349.9 200 253 320 

Engine specific impulse at altitude 462 445.6 290 288 320 

Engine mixture ratio (oxidizer/fuel) 5.88 6.0 2.0 2.27 1.90 



276 THRUST CHAMBERS 



8.1. INJECTORS 

The functions of an injector are similar to those of a carburetor of an internal 
combustion engine. The injector has to introduce and meter the flow of liquid 
propellants to the combustion chamber, cause the liquids to be broken up into 
small droplets (a process called atomization), and distribute and mix the propel- 
lants in such a manner that a correctly proportioned mixture of fuel and oxidizer 
will result, with uniform propellant mass flow and composition over the chamber 
cross section. 

There are two design approaches for admitting propellants into the combustion 
chamber. The individual locations on the injector face are called injector elements. 
The older types use a set of propellant jets going through a set of holes in a pattern 
on the injector face. Different hole arrangements are shown in Figure 8-3. Many 
of the rocket injectors developed in the United States used this type for large 
and small thrust chambers. The second type has cylindrical injection elements, 
which are inserted and fastened (welded, brazed, or soldered) into the injector 
face and each element delivers a conically shaped spray of propellants into the 
combustion chamber. Figure 8-4 shows several spray injection elements with 
conical sheets of propellant coming from a slot or from the internal edge of a 
hollow cylinder in the injection element. It has been used with LOX/LH 2 thrust 
chamber worldwide including the Space Shuttle thrust chamber. Also it has been 
the preferred approach in Russia (and its predecessor the Soviet Union), where 
it was used with most propellants and all thrust chamber sizes. There are also 
some combination types that use jets and sprays together. Typical injectors with 
hole patterns can be seen in Figs. 8-1 and 8-5. 

The injection pattern on the face of the injector is closely related to the 
internal manifolds or feed passages within the injector. These provide for the 
distribution of the propellant from the injector inlet to all the injection holes or 
spray elements. A large complex manifold volume allows low passage velocities 
and good distribution of flow over the cross section of the chamber. A small 
manifold volume allows for a lighter weight injector, a faster start, and reduces 
the amount of “dribble” flow after the main valves are shut. The higher passage 
velocities usually cause a more uneven flow through different identical injection 
holes and thus a poorer distribution and wider local gas composition variation. 
Dribbling results in after-burning, which is an inefficient irregular combustion 
that gives a little “cutoff’ thrust after valve closing. For applications with very 
accurate terminal vehicle velocity requirements, the cutoff impulse has to be very 
small and reproducible and often valves are built into the injector to minimize 
passage volume. 

Doublet impinging-stream-type, multiple-hole injectors are commonly used 
with oxygen-hydrocarbon and storable propellants, as seen in Fig. 8-3. For 
unlike doublet patterns the propellants are injected through a number of separate 
small holes in such a manner that the fuel and oxidizer streams impinge upon 
each other. Impingement forms thin liquid fans and aids atomization of the liquids 
into droplets, also aiding distribution. Characteristics of specific injector orifices 



8.1. INJECTORS 277 



Injection holes 



Impingment 

points 




Doublet impinging 
stream pattern 




Fuel 

manifolds 



Impingment 
point 



Face of 
injector 



Triplet impinging 
stream pattern 




Typical impingement 
point 



yA 

Seif- impinging 
stream pattern 




Typical straight 
propellant stream 



Fuel 
manifold^N^ 



Oxidizer 

manifold 

J 



^Premix chamber 

Injector lace 




' Oxidizer inlet 
^Fuel inlet 



.Conical splash 
plate 



Splash plate 
pattern 



Premixing type 

FIGURE 8-3. The upper four simplified sketches show common injection designs using 
holes. The premix chamber-type (its igniter is not shown) is used today on a few large 
LOX/LH 2 thrust chambers. The splash plate-type was popular in the late 1940s and 1950s 
but is not used today. (Used with permission from Ref. 8-1.) 



are given in Table 8-2. Impinging hole injectors are also used for like-on-like or 
self-impinging patterns (fuel-on-fuel and oxidizer-on-oxidizer). The two liquid 
streams then form a fan whieh breaks up into droplets. Unlike doublets work 
best when the hole size (more exactly, the volume flow) of the fuel is about 
equal to that of the oxidizer and the ignition delay is long enough to allow the 
formation of fans. For uneven volume flow the triplet pattern often seems to be 
more effective. 

The nonimpinging or shower head injector employs nonimpinging streams 
of propellant usually emerging normal to the face of the injector. It relies on 
turbulence and diffusion to achieve mixing. The German World War II V-2 
rocket used this type of injector. This type is no longer used because it requires 
a large chamber volume for good combustion. 



278 THRUST CHAMBERS 




Injeaor face 



(a) Single propellent spray element 
%vitn spiral Insert. 




Tangential ' 
Injeaion v 
orifices \\ 
(to make \\ 
flow swirl) V 







Iq. I’” 




li 




(cf) Coaxial blpropellant 

injection spray element with 
external burning. 




. Conical 
X spray 



Injeaor face 

(b)Slngle propellant spray element 
with tangential injection. 



Oxidizer , _ , 

Manifold^ I 



Tangential 
injection ^ 
orifices \\ 
(to make \\ 
flow swirl) V 




- 

^ impingment 
outside 



'' Injector 
(e) Coaxial blpropellant face 
injection sprayelement 
with Internal burning. 




-Cone spray 
without 
prerotation 
will break up 
Into small 
droplets 



(c) Pintle injection element 
for single propellant with 
fixed position pintle. 



Dual valve 
seats for 
shutoff 




Injector face' clings to 
(0 Coaxial bipropellant 
pintle element with 
movable pintle (throttling) 
and dual-face valves (for shutoff). 



FIGURE 8-4. Simplified cross-section sketches of several common types of spray injec- 
tion elements. (Used with permission from Ref. 8-1.) 



Sheet or spray-type injectors give cylindrical, conical, or other types of spray 
sheets; these sprays generally intersect and thereby promote mixing and atom- 
ization. See Fig. 8-4. The conical sheets break up into droplets, which are then 
vaporized. The size distribution of droplets from spray injection elements is usu- 
ally more uniform than with impinging streams from holes. By changing some 
of the internal dimensions of the spray elements (change size or number of tan- 
gential feed holes, the length/protrusion of an internal cylinder, or the angle of 
an internal spiral rib), it is possible to change the angle of the conical sheet or 
the location of the impingement of fuel and oxidizer spray sheets, and to affect 
the combustion efficiency. By varying the width of the sheet (through an axi- 
ally movable sleeve), it is possible to throttle the propellant flow over a wide 
range without excessive reduction in injector pressure drop. This type of variable 
area concentric tube injector was used on the descent engine of the Apollo lunar 
excursion module and throttled over a 10:1 range of flow with only a very small 
change in mixture ratio or performance. 



8.1. INJECTORS 279 




FIGURE 8-5. Injector with 90° self-impinging (fuel-against-fuel and oxidizer-against- 
oxidizer)-type countersunk doublet injection pattern. Large holes are inlets to fuel mani- 
folds. Predrilled rings are brazed alternately over an annular fuel manifold or groove and 
a similar adjacent oxidizer manifold or groove. A section through a similar but larger 
injector is shown in Fig. 8-1. 



The coaxial hollow post or spray injector has heen used for liquid oxygen 
and gaseous hydrogen injectors hy most domestic and foreign rocket designers. 
It is shown in sketches d and e of Fig. 8-4. It works well when the liquid 
hydrogen has absorbed heat from cooling jackets and has been gasified. This 
gasified hydrogen flows at high speed (typically 330 m/sec or 1000 ft/sec); the 
liquid oxygen flows far more slowly (usually at less than 33 m/sec or 100 ft/sec), 
and the differential velocity causes a shear action, which helps to break up the 
oxygen stream into small droplets. The injector has a multiplicity of these coaxial 
elements on its face. In Russia and in Germany the spray injector elements have 
also been used with storable propellants. 

The SSME injector shown in Fig. 9-6 uses 600 of these concentric sleeve 
injection elements; 75 of them have been lengthened beyond the injector face to 
form cooled baffles, which reduce the incidence of combustion instability. 

The original method of making injection holes was to carefully drill them and 
round out or chamfer their inlets. This is still being done today. It is difficult 
to align these holes accurately (for good impingement) and to avoid burrs and 
surface irregularities. One method that avoids these problems and allows a large 
number of small accurate injecton orifices is to use multiple etched, very thin 
plates (often called platelets) that are then stacked and diffusion bonded together 




280 THRUST CHAMBERS 



TABLE 8-2. Injector Discharge Coefficients 



Orifice Type 



Discharge 

Diagram Diameter (mm) Coefficient 



Sharp-edged orifice 




Above 2.5 0.61 

Below 2.5 0.65 approx. 



Short-tube with rounded 
entrance L/D >3.0 




1.00 0.88 

1.57 0.90 

1.00 

(with L/D ~ 1.0 ) 0.70 



Short tube with conical 
entrance 




0.50 0.7 

1.00 0.82 

1.57 0.76 

2.54 0.84-0.80 

3.18 0.84-0.78 



Short tube with spiral effect 




1.0-6.4 0.2-0.55 



Sharp-edged cone 




1.00 0.70-0.69 

1.57 0.72 



to form a monolithic structure as shown in Fig. 8-6. The photo-etched pattern 
on each of the individual plates or metal sheets then provides not only for many 
small injection orifices at the injector face but also for internal distribution or 
flow passages in the injector and sometimes also for a fine-mesh filter inside the 
injector body. The platelets can be stacked parallel to or normal to the injector 
face. The finished injector has been called the platelet injector and has been 
patented by the Aerojet Propulsion Company. 

Injector Flow Characteristics 

The differences of the various injector element configurations shown in Figs. 8-3 
and 8-4 reflect themselves in different hydraulic flow-pressure relationships, dif- 
ferent starting characteristics, atomization, resistance to self-induced vibrations, 
and combustion efficiency. 

The hydraulic injector characteristics can be evaluated accurately and can 
be designed for orifices with the desired injection pressures, injection velocities, 
flows, and mixture ratio. For a given thrust F and a given effective exhaust 



8.1. INJECTORS 281 




Face plate with 
inclined injection holes 
(laser drilled) 




Fuel distribution 
plate with etched 
cooling passages 




Oxidizer distribution 
manifold plate 



Alignment pins (2) 




Cover plate with 
oxidizer inlet 




Four stacked plates 
(with exaggerated thickness) 



Oxidizer filter 
Fuel manifold 



Oxidizer manifold 

Oxidizer 
inlet port 




\ 

Intergrat 

Like on like 
impinging orifices 

Cross section through an etched plate 



(h) 




Enlarged detail segment of 
one design of like-on-like impinging 
orifices and feed passage geometry 



FIGURE 8-6. Simplified diagrams of two types of injector using a bonded platelet 
construction technique: (a) injector for low thrust with four impinging unlike doublet 
liquid streams; the individual plates are parallel to the injector face; (b) like-on-like 
impinging stream injector with 144 orifices; plates are perpendicular to the injector face. 
(Courtesy of Aerojet Propulsion Company). 



velocity c, the total propellant mass flow m is given by m = F fc from Eq. 2-6. 
The relations between the mixture ratio, the oxidizer, and the fuel flow rates 
are given by Eqs. 6-1 to 6-4. For the of an incompressible fluid through 
hydraulic orifices, 



Q = CdA^l^pjp 



( 8 - 1 ) 




282 THRUST CHAMBERS 



m = Qp = CdA^/lfTKp (8-2) 

where Q is the volume flow rate, Q the dimensionless discharge coefficient, 
p the propellant mass density, A the cross-sectional area of the orifice, and Ap 
the pressure drop. These relationships are general and can be applied to any one 
section of the propellant feed system, to the injector, or to the overall liquid flow 
system. 

For any given pressure drops the injection orifices usually determine the mix- 
ture ratio and the propellant flows of the rocket unit. From Eqs. 6-1 and 8-2 
the mixture ratio r is 

r = tho/mf = [iCd)o/{Cd)f]iAo/Af)y/ipo/pf){Apo/Apf) (8-3) 

The quantities in the preceding equations have to be chosen so that the correct 
design mixture ratio is attained. Orifices whose discharge coefficients are constant 
over a large range of Reynolds numbers and whose ratio {Cd)o/{Cd)f remains 
essentially invariant should be selected. For a given injector it is usually difficult 
to maintain the mixture ratio constant at low flows or thrusts, such as in starting. 

The quality of the injector can be checked by performing cold tests with 
inert simulant liquids instead of reactive propellant liquids. Often water is used 
to confirm pressure drops through the fuel or oxidizer side at different flows, 
and this allows determination of the pressure drops with propellants and the 
discharge coefficients. Nonmixable inert liquids are used with a special apparatus 
to determine the local cold flow mixture ratio distribution over the chamber cross 
section. The simulant liquid should be of approximately the same density and 
viscosity as the actual propellant. New injectors are usually hot fired and tested 
with actual propellants. 

The actual mixture ratio can be estimated from cold flow test data, the mea- 
sured hole areas, and discharge coefficients by correcting by the square root of the 
density ratio of the simulant liquid and the propellant. When water at the same 
pressure is fed alternately into both the fuel and the oxidizer sides, Apf = Ap„ 
and pf = po and the water mixture ratio will be 

r = [{Cd)o/{Cd)f]AJAf (8-4) 

Therefore, the mixture ratio measured in water tests can be converted into the 
actual propellant mixture ratio by multiplying it by the square root of the density 
ratio of the propellant combination and the square root of the pressure drop ratio. 
The mechanism of propellant atomization with simultaneous vaporization, partial 
combustion, and mixing is difficult to analyze, and performance of injectors has to 
be evaluated by experiment within a burning rocket thrust chamber. The injection 
velocity is given by 



V = Q/A = CdyjlApjp 



(8-5) 



8.1. INJECTORS 283 



Values of discharge coefficients for various types of injection orifices are shown 
in Table 8-2. The velocity is a maximum for a given injection pressure drop when 
the discharge coefficient approaches 1 . Smooth and well-rounded entrances to the 
injection holes and clean bores give high values of the discharge coefficient, and 
this hole entry design is the most common. Small differences in chamfers, hole 
entry radius, or burrs at the edge of a hole can cause significant variations in 
the discharge coefficient and the jet flow patterns, and these in turn can alter the 
quality and distribution of the atomized small droplets, the local mixture ratio, 
and the local heat transfer rates. An improperly manufactured hole can cause 
local chamber or injector burnout. 

When an oxidizer and a fuel jet impinge, the resultant momentum can be 
calculated from the following relation, based on the principle of conservation of 
momentum. Figure 8-7 illustrates a pair of impinging jets and defines j/q the 
angle between the chamber axis and the oxidizer stream, y/ as the angle between 
the chamber axis and the fuel stream, and 8 as the angle between the chamber 
axis and the average resultant stream. If the total momentum of the two jets 
before and after impingement is equal. 



tan 8 = 



nioVo sin /o — tn/Vf sin yy 
thoVo cos Yo + thfVf cos yy 



( 8 - 6 ) 



Good performance is often obtained when the resultant momentum of impinging 
streams is approximately axial. If the resultant momentum is along the chamber 
axis, 8 = 0, tan 5 = 0, and the angular relation for an axially directed jet momen- 
tum is given by 



mono sin yo = rhfVf sin yy (8-7) 

From these equations the relation between yf,yo, and 8 can be determined. A 
sample injector analysis is shown in Section 8.8. 



Line of resultant 
jet momentum 





FIGURE 8-7. Angular relation of doublet impinging-stream injection pattern. 



284 THRUST CHAMBERS 



Factors Influencing Injector Behavior 

A complete theory relating injector design parameters to rocket performance and 
combustion phenomena has not yet been devised, and therefore the approach to 
the design and development of liquid propellant rocket injectors has been largely 
empirical. Some of the analyses (see Ref. 8-2) has been useful in understanding 
the phenomenology and in indicating the directions for injector development. The 
available data indicate several important factors that affect the performance and 
operating characteristics of injectors; some of these are briefly enumerated here. 
These factors are different for injectors where both propellants are liquid (such 
as nitrogen tetroxide and hydrazine) or where one propellant is liquid and the 
other is gaseous as with LOX/gasified liquid hydrogen. Monopropellant injectors 
feed propellants into a catalyst bed and they are different and described in 
Section 8.3. 



Propellant Combination. The particular combination of fuel and oxidizer 
affects such characteristics as the relative chemical reactivity, the ease and speed 
of vaporization, the ignition temperature, the diffusion of hot gases, the volatil- 
ity, or the surface tension. Hypergolic (self-igniting) propellants generally require 
injector designs somewhat different from those required by propellants that must 
be ignited. Injector designs that perform efficiently with one combination do not 
necessarily work too well with a different propellant combination. 

Injection, Element Pattern, and Orifice Size. With individual elements of 
holes or sprays in the injector plate, there appears to be an optimum performance 
and/or heat transfer condition for each of the following parameters: orifice size, 
angle of impingement, angle of resultant momentum, distance of the impingement 
locus from the injector face, number of injection orifices per unit of injector face 
surface, flow per unit of injection orifice, and distribution of orifices over the 
injector face. These parameters are largely determined experimentally or from 
similar earlier successful injectors. 



Transient Conditions. Starting and stopping may require special provisions 
(temporary plugging of holes, accurate valve timing, insertion of paper cups over 
holes to prevent entry of one propellant into the manifold of the other propellant 
as was done on the German A-4 or V-2 thrust chamber, or check valves) to 
permit satisfactory transient operation. 



Hydraulic Characteristics. The orifice type and the pressure drop across the 
injection orifice determine the injection velocity. A low pressure drop is desirable 
to minimize the weight of the feed system or the pumping power. High-pressure 
drops are used often to increase the rocket’s resistance to combustion instability 
and enhance atomization of the liquids and improve performance. 



8.2. COMBUSTION CHAMBER AND NOZZLE 285 



Heat Transfer. Injectors influence the performance and the heat transferred 
in rocket thrust chambers. Low heat transfer rates have been obtained when 
the injection pattern resulted in an intentionally rich mixture near the chamber 
walls and when the chamber pressure is low. In general, the higher performance 
injectors have a higher heat transfer rate to the walls of the combustion chamber, 
the nozzle, and the injector face. 

Structural Design. The injector is highly loaded by pressure forces from the 
combustion chamber and the propellant manifolds. During transition (starting or 
stopping) these pressure conditions can cause internal transient structural stresses 
which sometimes exceed the steady-state operating conditions. The faces of most 
modem injectors are flat and must be reinforced by suitable structures which 
nevertheless should provide no obstmctions to the hydraulic manifold passages; 
the structure must also be sufficiently flexible to allow thermal deformations 
caused by heating the injector face with hot combustion gases or cooling of 
certain flow passages by cryogenic propellants. The injector design must also 
provide for positive seals between fuel and oxidizer manifolds (an internal leak 
can cause manifold explosions or internal fires) and a sealed attachment of the 
injector to the chamber. In large, gimbal-mounted thmst chambers the injector 
also often carries the main thrust load, and a gimbal mount is often directly 
attached to the injector, as shown in Figs. 6-1 and 8-1. 

Combustion Stability. The injection hole or spray pattern, impingement pat- 
tern, hole or spray element distribution, and pressure drop have a strong influence 
on combustion stability; some types are much more resistant to pressure distur- 
bances. As explained in Section 9.3, the resistance to vibration is determined 
experimentally, and often special antivibration devices, such as baffles or reso- 
nance cavities, are designed directly into the injeetor. 



8.2. COMBUSTION CHAMBER AND NOZZLE 

The combustion chamber is that part of a thrust chamber where the eombustion 
or burning of the propellant takes place. The combustion temperature is much 
higher than the melting points of most chamber wall materials. Therefore it is 
necessary either to cool these walls (as described in a later section of this chapter) 
or to stop rocket operation before the critical wall areas become too hot. If the 
heat transfer is too high and thus the wall temperatures become locally too high, 
the thrust chamber will fail. Heat transfer to thrust chambers will be described 
later in this chapter. Section 8.8 gives a sample analysis of a thrust chamber and 
Ref. 8-2 describes the analyses for design and development. 

Volume and Shape 

Spherical chambers give the least internal surface area and inert mass per unit 
chamber volume; they are expensive to build and several have been tried. Today 



286 THRUST CHAMBERS 



we prefer a cylindrical chamber (or slightly tapered cone frustum) with a flat 
injector and a converging -diverging nozzle. The chamber volume is defined as 
the volume up to the nozzle throat section, and it includes the cylindrical chamber 
and the converging cone frustum of the nozzle. Neglecting the effect of the corner 
radii, the chamber volume Vc is 

y, = AiLi +AiL,(l + VA,/Ai +A,Mi) (8-8) 

Here Li is the cylinder length, At/Ai is the chamber contraction ratio, and Lc is 
the length of the conical frustum. The chamber surfaces exposed to heat transfer 
from hot gas are the injector face, the inner surface of the cylinder chamber, and 
the inner surface of the converging cone frustrum. The volume and shape are 
selected after evaluating these parameters: 

1 . The volume has to be large enough for adequate atomization, mixing, evapo- 
ration, and complete combustion of propellants. Chamber volumes vary for 
different propellants with the time delay necessary to vaporize and activate 
the propellants and with the speed of combustion reaction of the propel- 
lant combination. When the chamber volume is too small, combustion is 
incomplete and the performance is poor. With higher chamber pressures 
or with highly reactive propellants, and with injectors that give improved 
mixing, a smaller chamber volume is usually permissible. 

2. The chamber diameter and volume can influence the cooling requirements . 
If the chamber volume and the chamber diameter are large, the heat transfer 
rates to the walls will be reduced, the area exposed to heat will be large, and 
the walls are somewhat thicker. Conversely, if the volume and cross section 
are small, the inner wall surface area and the inert mass will be smaller, but 
the chamber gas velocities and the heat transfer rates will be increased. For 
the same thrust chamber there is an optimum chamber volume and diameter 
where the total heat absorbed by the walls will be a minimum. This is 
important when the available cooling capacity of the coolant is limited (e.g., 
oxygen-hydrocarbon at high mixture ratios) or if the maximum permissive 
coolant temperature has to be limited (for safety reasons with hydrazine 
cooling). The total heat transfer can also be further reduced by going to a 
rich mixture ratio or by adding film cooling (discussed below). 

3. All inert components should have minimum mass. The thrust chamber mass 
is a function of the chamber dimensions, chamber pressure, and nozzle area 
ratio, and the method of cooling. 

4. Manufacturing considerations favor a simple chamber geometry, such as 
a cylinder with a double cone bow-tie-shaped nozzle, low-cost materials, 
and simple fabrication processes. 

5. In some applications the length of the chamber and the nozzle relate directly 
to the overall length of the vehicle. A large-diameter but short chamber 
and/or a short nozzle can allow a somewhat shorter vehicle with a lower 
structural inert vehicle mass. 



8.2. COMBUSTION CHAMBER AND NOZZLE 287 



6. The gas pressure drop for accelerating the combustion products within the 
chamber should be a minimum; any pressure reduction at the nozzle inlet 
reduces the exhaust velocity and the performance of the vehicle. These 
losses become appreciable when the chamber area is less than three times 
the throat area. 

7. For the same thrust the combustion volume and the nozzle throat area 
become smaller as the operating chamber pressure is increased. This means 
that the chamber length and the nozzle length (for the same area ratio) also 
decrease with increasing chamber pressure. The performance and the heat 
transfer also go up with chamber pressure. 

The preceding chamber considerations conflict with each other. It is, for 
instance, impossible to have a large chamber that gives complete combustion 
but has a low mass. Depending on the application, a compromise solution that 
will satisfy the majority of these considerations is therefore usually selected and 
verified by experiment. 

The characteristic chamber length is defined as the length that a chamber of 
the same volume would have if it were a straight tube and had no converging 
nozzle section: 



L* = Vc/A, (8-9) 

where L* (pronounced el star) is the characteristic chamber length. At is the 
nozzle throat area, and Vc is the chamber volume. The chamber includes all the 
volume up to the throat area. Typical values for L* are between 0.8 and 3.0 m (2.6 
to 10 ft) for several bipropellants and higher for some monopropellants. Because 
this parameter does not consider any variables except the throat area, it is useful 
only for a particular propellant combination and a narrow range of mixture ratio 
and chamber pressure. The parameter L* was used about 40 years ago, but today 
the chamber volume and shape are chosen by using data from successful thrust 
chambers of prior similar designs and identical propellants. 

The stay time t, of the propellant gases is the average value of the time spent 
by each molecule or atom within the chamber volume. It is defined by 



f, = Vc/{mVi) (8-10) 

where m is the propellant mass flow, Vi is the average specific volume or volume 
per unit mass of propellant gases in the chamber, and Vc is the chamber vol- 
ume. The minimum stay time at which a good performance is attained defines the 
chamber volume that gives essentially complete combustion. The stay time varies 
for different propellants and has to be experimentally determined. It includes the 
time necessary for vaporization, activation, and complete burning of the propel- 
lant. Stay times have values of 0.001 to 0.040 sec for different types and sizes 
of thrust chambers and for different propellants. 



288 THRUST CHAMBERS 



The nozzle dimensions and configuration can be determined from the analyses 
presented in Chapter 3. The converging section of the nozzle experiences a much 
higher internal gas pressure than the diverging section and therefore the design 
of the converging wall is similar to the design of the cylindrical chamber wall. 
The exact contour of the converging nozzle section is not critical. Most thrust 
chambers use a shortened bell shape for the diverging nozzle section. Nozzles 
with area ratios over 400 have been developed and flown. 

In Chapter 3 it was stated that very large nozzle exit area ratios allow a 
small but significant improvement in specific impulse, particularly at very high 
altitudes; however, the extra length, extra nozzle mass, and extra vehicle mass 
necessary to house a large nozzle often make this unattractive. This disadvantage 
can be mitigated by a multipiece nozzle that is stored in annular pieces around 
the engine during the ascent of the launch vehicle and automatically assem- 
bled in space after launch vehicle separation and before firing. This concept, 
known as extendible nozzle, has been successfully employed in solid propellant 
rocket motors for space applications for about 27 years. The first flight with an 
extendible nozzle on a liquid propellant engine was performed in 1998 with a 
modified version of a Pratt & Whitney Rocketdyne upper stage engine. Its flight 
performance is listed in Table 8-1. The engine is shown later in Fig. 8-17 and 
its carbon-carbon extendible nozzle cone is described in the section on materials 
and fabrication of this chapter. 



Heat Transfer Distribution 

Heat is transmitted to all internal hardware surfaces exposed to hot gases, namely 
the injector face, the internal chamber and nozzle walls. The heat transfer rate 
or heat transfer intensity, that is, local wall temperatures and heat transfer per 
unit wall area, varies within the thrust chamber. A typical heat transfer rate 
distribution is shown in Fig. 8-8. Only 0.5 to 5% of the total energy generated 
in the gas is transferred as heat to the chamber walls. For a typical rocket of 
44,820 N or 10,000 Ibf thrust the heat rejection rate to the wall may be between 
0.75 and 3.5 MW, depending on the exact conditions and design. See Section 8.5. 

The amount of heat transferred by conduction from the chamber gas to the 
walls in a rocket thrust chamber is negligible. By far the largest part of the heat is 
transferred by means of convection. A part (usually 5 to 35%) of the transferred 
heat is attributable to radiation. 

For constant chamber pressure, the chamber wall surface increases less rapidly 
than the volume as the thrust level is raised. Thus the cooling of chambers is 
generally easier in large thrust sizes, and the capacity of the wall material or the 
coolant to absorb all the heat rejected by the hot gas is generally more critical 
in smaller sizes because of the volume -surface relationship. 

Higher chamber pressure leads to higher vehicle performance (higher /j), but 
also to higher engine inert mass. However, the resulting increase of heat transfer 
with chamber pressure often imposes design or material limits on the maximum 
practical chamber pressure for both liquid and solid propellant rockets. 



8.2. COMBUSTION CHAMBER AND NOZZLE 



289 




FIGURE 8-8. Typical axial heat transfer rate distribution for liquid propellant thrust 
chambers and solid propellant rocket motors. The peak is always at the nozzle throat or 
slightly upstream of the throat and the lowest value is usually near the nozzle exit. 



The heat transfer intensity in chemical rocket propulsion can vary from less 
than 50 W/cm^ or 0.3 Btu/in.^-sec to over 16 kW/cm^ or 100 Btu/in.^-sec. The 
high values are for the nozzle throat region of large bipropellant thrust chambers 
and high-pressure solid rocket motors. The lower values are for gas generators, 
nozzle exit sections, or small thrust chambers at low chamber pressures. 

Cooling of Thrust Chambers 

The primary objective of cooling is to prevent the chamber and nozzle walls 
from becoming too hot, so they will no longer be able to withstand the imposed 
loads or stresses, thus causing the chamber or nozzle to fail. Most wall materials 
lose strength and become weaker as temperature is increased. These loads and 
stresses are discussed in the next section. With further heating, the walls would 
ultimately fail or even melt. Cooling thus reduces the wall temperatures to an 
acceptable value. 

Basically, there are two cooling methods in common use today. The first is 
the steady-state method. The heat transfer rate and the temperatures of the cham- 
bers reach thermal equilibrium. This includes regenerative cooling and radiation 
cooling. The duration is limited only by the available supply of propellant. 

Regenerative cooling is done by building a cooling jacket around the thrust 
chamber and circulating one of the liquid propellants (usually the fuel) through 
it before it is fed to the injector. This cooling technique is used primarily with 
bipropellant chambers of medium to large thrust. It has been effective in applica- 
tions with high chamber pressure and high heat transfer rates. Also, most injectors 
use regenerative cooling on their hot faces. 

In radiation cooling the chamber and/or nozzle have only a single wall made 
of high-temperature material, such as niobium or molybdenum. When it reaches 



290 THRUST CHAMBERS 



thermal equilibrium, this wall usually glows red or white hot and radiates heat 
away to the surroundings or to empty space. Radiation cooling is used with 
bipropellant and monopropellant thrusters, bipropellant and monopropellant gas 
generators, and for diverging nozzle exhaust sections beyond an area ratio of 
about 6 to 10 in larger thrust chambers (see Fig. 8-2). A few small bipropellant 
thrusters are also radiation cooled. This cooling scheme has worked well with 
lower chamber pressures (less than 250 psi) and moderate heat transfer rates. 

The second cooling method relies on transient heat transfer or unsteady heat 
transfer. It is also called heat sink cooling. The thrust chamber does not reach a 
thermal equilibrium, and temperatures continue to increase with operating dura- 
tion. The heat-absorbing capacity of the metal hardware determines its maximum 
duration, which is usually short (a few seconds for an all-metal construction). 
The rocket combustion operation has to be stopped just before any of the exposed 
walls reaches a critical temperature at which it could fail. This method has mostly 
been used with low chamber pressures and low heat transfer rates. Heat sink cool- 
ing of thrust chambers can also be done by absorbing heat in an inner liner made 
of an ablative material, such as hber-reinforced plastics. Ablative materials burn 
or erode slowly and cumulative operating durations can be minutes. Ablative 
materials are used extensively in solid propellant rocket motors and will be dis- 
cussed further in Chapter 15. The analysis of both of these cooling methods is 
given in the next section of this chapter. 

Film cooling and special insulation are supplementary techniques that are used 
occasionally with both methods to locally augment their cooling capability. All 
these cooling methods will be described further in this chapter. 

Cooling also helps to reduce the oxidation of the wall material and the rate 
at which walls would be eaten away. The rates of chemical oxidizing reactions 
between the hot gas and the wall material can increase dramatically with wall 
temperature. This oxidation problem can be minimized not only by limiting the 
wall temperature but also by burning the liquid propellants at a mixture ratio 
where the percentage of aggressive gases in the hot gas (such as oxygen or 
hydroxyl) is very small, and by coating certain wall materials with an oxidation- 
resistant coating; for example, iridium has been coated on the inside of rhenium 
walls. 

Cooling with Steady-State Heat Transfer. Cooled thrust chambers have 
provisions for cooling some or all metal parts coming into contact with hot gases, 
such as chamber walls, nozzle walls, and injector faces. Internal cooling passages, 
cooling jackets, or cooling coils permit the circulation of a coolant. Jackets can 
consist of separate inner and outer walls or of an assembly of contoured, adjacent 
tubes (see Figs. 8-1 and 8-9). The inner wall confines the gases, and the spaces 
between the walls serve as the coolant passage. The nozzle throat region is 
usually the location that has the highest heat transfer intensity and is therefore 
the most difficult to cool. For this reason the cooling jacket is often designed 
so that the coolant velocity is highest at the critical regions by restricting the 
coolant passage cross section, and so that the fresh cold coolant enters the jacket 



8.2. COMBUSTION CHAMBER AND NOZZLE 291 




Top view without 
manifold 



Exit (Section C) 



ocnc 

WO 



Chamber 
(Section B) 
Throat 
(Section A) 



FIGURE 8-9. Diagram of a tubular cooling jacket. The tubes are bent to the chamber 
and nozzle contours; they are formed hydraulically to give a variable cross section to 
permit the same number of tubes at the throat and exit diameters. Coolant enters through 
the inlet manifold into every other tube and proceeds axially to the nozzle exit manifold, 
where it then enters the alternate tubes and returns axially to go directly to the injector. 



at or near the nozzle. While the selection of the coolant velocity and its variation 
along the wall for any given thrust chamber design depends on heat transfer 
considerations, the selection of the coolant passage geometry often depends on 
pressure loss, stresses, and manufacturing considerations. An axial flow cooling 
jacket, or a tubular wall, has a low hydraulic friction loss but is practical only 
for large coolant flows (above approximately 9 kg/sec). For small coolant flows 
and small thrust units, the design tolerances of the cooling jacket width between 
the inner and outer walls or the diameters of the tubes, become too small, or 
the tolerances become prohibitive. Therefore, most small thrust chambers use 
radiation cooling or ablative materials. 

In regenerative cooling the heat absorbed by the coolant is not wasted; it 
augments the initial energy content of the propellant prior to injection, increasing 
the exhaust velocity slightly (0.1 to 1.5%). This method is called regenerative 
cooling because of the similarity to steam regenerators. The design of the tubular 
chamber and nozzle combines the advantages of a thin wall (good for reducing 
thermal stresses and high wall temperatures) and a cool, lightweight structure. 
Tubes are formed to special shapes and contours (see Figs. 8-1 and 8-9), usually 
by hydraulic means, and then brazed, welded, or soldered together (see Ref. 8-3). 
In order to take the gas pressure loads in hoop tension, they are reinforced on 
the outside by high-strength bands or wires. While Fig. 8-9 shows alternate 
tubes for up and down flow, there are chambers where the fuel inlet manifold 
is downstream of the nozzle throat area and where the coolant flow is up and 
down in the nozzle exit region, but unidirectionally up in the throat and chamber 
regions. 

Radiation cooling is another steady-state method of cooling. It is simple and 
is used extensively in the low heat transfer applications listed previously. Fur- 
ther discussion of radiation cooling is given in Section 8.4. In order for heat to 
be radiated into space, it is usually necessary for the bare nozzle and chamber 



292 THRUST CHAMBERS 



to Stick out of the vehicle. Figure 8-16 shows a radiation-cooled thrust cham- 
ber. Since the white hot glowing radiation-cooled chambers and/or nozzles are 
potent radiators, they may cause undesirable heating of adjacent vehicle or engine 
components. Therefore, many have insulation (see Fig. 8-13) or simple exter- 
nal radiation shields to minimize these thermal effects; however, in these cases 
the actual chamber or nozzle wall temperatures are higher than they would be 
without the insulation or shielding. 

Cooling with Transient Heat Transfer. Thrust chambers with unsteady heat 
transfer are basically of two types. One is a simple metal chamber (steel, copper, 
stainless steel, etc.) made with walls sufficiently thick to absorb plenty of heat 
energy. For short-duration testing of injectors, testing of new propellants, rating 
combustion stability, and very short duration rocket-propelled missiles, such as 
an antitank weapon, a heavy-walled simple, short-duration steel chamber is often 
used. The common method of ablative cooling or heat sink cooling uses a combi- 
nation of endothermic reactions (breakdown or distillation of matrix material into 
smaller compounds and gases), pyrolysis of organic materials, counter-current 
heat flow and coolant gas mass flow, charring, and localized melting. An abla- 
tive material usually consists of a series of strong, oriented fibers (such as glass, 
Kevlar, or carbon fibers) engulfed by a matrix of an organic binder material (such 
as plastics, epoxy resins or phenolic resins). As shown in Fig. 15-11, the gases 
seep out of the matrix and form a protective film cooling layer on the inner wall 
surfaees. The fibers and the residues of the matrix form a hard char or porous 
coke-like material that preserves the wall contour shapes. 

The orientation, number, and type of fiber determine the ability of the com- 
posite ablative material to withstand significant stresses in preferred directions. 
For example, internal pressure produces longitudinal as well as hoop stresses in 
the thrust chamber walls and thermal stresses produce compression on the inside 
of the walls and tensile stresses on the outside. We have learned how to place the 
fibers in two or three directions, which makes them anisotropic. We then speak 
of two-dimensional (2-D) and 3-D fiber orientation. 

A set of strong carbon fibers in a matrix of amorphous carbon is a special, 
but favorite type of material. It is often abbreviated as C-C or carbon-carbon. 
The carbon materials lose their ability to carry loads at a high temperature of 
about 3700 K or 6200 F. Carbon oxidizes readily form CO or CO 2 . Its best 
applications are with fuel-rich propellant mixtures that have little or no free 
oxygen or hydroxyl in their exhaust. It has also been used for nozzle throat 
inserts. Properties for one type of C-C are given in Table 15-5. AC-C nozzle 
extension is shown in Fig. 8-17. See Ref. 8-4. 

Ablative cooling was first used and is still used extensively with solid pro- 
pellant rocket motors. It has since been successfully applied to liquid propel- 
lant thrust chambers, particularly at low chamber pressure, where the static gas 
temperatures are relatively low. It is still used today as a material for nozzle 
extensions, such as for the RS-68 in Fig. 6-10, and it can operate for several 
minutes. It is also used as a chamber and nozzle liner at low chamber pres- 
sure. An example is the axial gimbaled thruster of the Peacekeeper fourth stage 



8.2. COMBUSTION CHAMBER AND NOZZLE 293 



which is seen in Figure 6-13. Ablative lined small thrusters (100 Ibf thrust or 
less) have been flown extensively in the 1950s and 1960s in Apollo missions 
and other applications for attitude control and minor maneuvers. They are no 
longer used today because they are relatively heavy and because the eroded par- 
ticle, droplets, or certain gases in the exhaust plume deposit condense on optical 
surfaces of spacecraft (mirror, solar cells, or windows). 

It is often advantageous to use a different cooling method for the downstream 
part of the diverging nozzle section, because its heat transfer rate per unit area 
is usually much lower than in the chamber or the converging nozzle section, 
particularly with nozzles of large area ratio. There is usually a small saving in 
inert engine mass, a small increase in performance, and a cost saving, if the 
chamber and the converging nozzle section and the throat region (up to an area 
ratio of perhaps 5 to 10) use regenerative cooling and the remainder of the nozzle 
exit section is radiation cooled (or sometimes ablative cooled). See Fig. 8-2 and 
Ref. 8-4. 

Film Cooling. This is an auxiliary method applied to chambers and/or nozzles, 
augmenting either a marginal steady-state or a transient cooling method. It can 
be applied to a complete thrust chamber or just to the nozzle, where heat transfer 
is the highest. Film cooling is a method of cooling whereby a relatively cool 
thin fluid film covers and protects exposed wall surfaces from excessive heat 
transfer. Figure 8-10 shows film-cooled chambers. The film can be introduced 
by injecting small quantities of fuel or an inert fluid at very low velocity through 
a large number of orifices along the exposed surfaces in such a manner that a 
protective relatively cool gas film or cool boundary layer is formed. In liquid 
propellant rocket engines extra fuel can also be admitted through extra injection 
holes at the outer layers of the injector; thus a propellant mixture is achieved (at 
the periphery of the chamber), which has a much lower combustion temperature. 
This differs from film cooling or transpiration cooling because there does not 
have to be a chamber cooling jacket or film-cooling manifolds. 

Film cooling by itself (without another cooling method) has been very effective 
in keeping chamber and nozzle materials from becoming too hot. In fact the very 
first thmst chambers developed by Robert H. Goddard in the 1920s were film 
cooled. However, the film coolant is not burned effectively and there was a 5 
to 17% reduction in specific impulse. Therefore, today, film cooling is used in 
small quantities (1 to 6% of fuel) to locally supplement other cooling methods 
and the performance loss is 0.5 to 2%. In solid propellant rocket engines this 
film cooling can be accomplished by inserting a ring of cool-burning propellant 
upstream of the nozzle, as shown in Fig. 8-10 or by wall insulation materials, 
whose ablation and charring will release relatively cool gases into the boundary 
layer. 

Turbine discharge gas (400 to 800°C or 752 to 1972°F) has also been used as 
a film coolant for uncooled nozzle exit sections of large liquid propellant rocket 
engines. Of course, the injection of an annular gas layer at the periphery of the 
nozzle exit, at a temperature lower than the maximum possible value, causes 



294 THRUST CHAMBERS 




FIGURE 8-10. Simplified diagrams of three different methods of forming a cool bound- 
ary layer in the nozzle. 



a decrease in a specific impulse. Therefore, it is desirable to reduce both the 
thickness of this cooler layer and the mass flow of cooler gas, relative to the 
total flow, to a practical minimum value at which cooling will still be effective. 

A special type of film cooling, sweat cooling or transpiration cooling, uses 
a porous wall material which admits a coolant through pores uniformly over the 
surface. This technique has been successfully used to cool injector faces in the 
upper stage engine (J-2) of the moon launch vehicle and the Space Shuttle main 
engine (SSME) with hydrogen fuel. 

Thermal Insulation. Theoretically, a good thermal insulation layer on the gas 
side of the chamber wall can be very effective in reducing chamber wall heat 
transfer and wall temperatures. However, efforts with good insulation materials 
such as refractory oxides or ceramic carbides have usually not been success- 
ful. They will not withstand differential thermal expansion without cracking or 
spalling. A sharp edge on the surface (crack or flaked-off piece of insulator) 
will cause a sudden rise in the stagnation temperature and most likely lead to 
a local wall failure. Asbestos is a good insulator and was used several decades 
ago; because it is a cancer-causing agent, it is no longer used. Coating develop- 
ment efforts with rhenium and other materials are continuing. Insulation or heat 
shields have been successfully applied on the exterior of radiation-cooled thrust 
chambers to reduce the heat transfer from the hot thrust chamber wall to adjacent 
sensitive equipment or structures. 

With hydrocarbon fuels it is possible to form small carbon particles or soot in 
the hot gas and that can lead to a carbon deposit on the gas side of the chamber 



8.2. COMBUSTION CHAMBER AND NOZZLE 295 



or nozzle walls. If it is a thin, mildly adhesive soot, it can be an insulator, but it 
is difficult to reproduce such a coating. More likely it forms hard, caked deposits, 
which can be spalled off in localized flakes and form sharp edges, and then it is 
undesirable. Most designers have preferred instead to use film cooling or extra 
high coolant velocities in the cooling jacket with injectors that do not create 
adhesive soot. 

Hydraulic Losses in the Cooiing Passage 

The cooling coil or cooling jacket should be designed so that the fluid adsorbs all 
the heat transferred across the inner motor wall, and so that the coolant pressure 
drop will be small. 

A higher pressure drop allows a higher coolant velocity in the cooling jacket, 
will do a better job of cooling, but requires a heavier feed system, which increases 
the engine mass slightly and thus also the total inert vehicle mass. For many 
liquid propellant rockets the coolant velocity in the chamber is approximately 
3 to 10 m/sec or 10 to 33 ft/sec and, at the nozzle throat, 6 to 24 m/sec or 20 to 
80 ft/sec. 

A cooling passage can be considered to be a hydraulic pipe, and the friction 
loss can be calculated accordingly. For a straight pipe, 

Ap/p^\fvHL/D) ( 8 - 11 ) 

where Ap is the friction pressure loss, p the coolant mass density, L the length of 
coolant passage, D the equivalent diameter, v the average velocity in the cooling 
passage, and / the friction loss coefficient. In English engineering units the right 
side of the equation has to be divided by go> the sea-level acceleration of gravity 
(32.2 ft/sec^). The friction loss coefficient is a function of Reynolds number and 
has values between 0.02 and 0.05. A typical pressure loss of a cooling jacket is 
between 5 and 25% of the chamber pressure. 

A large portion of the pressure drop in a cooling jacket usually occurs in those 
locations where the flow direction or the flow-passage cross section is changed. 
Here the sudden expansion or contraction causes a loss, sometimes larger than 
the velocity head u^/2. This hydraulic situation exists in inlet and outlet chamber 
manifolds, injector passages, valves, and expansion joints. 

The pressure loss in the cooling passages of a thrust chamber can be calculated, 
but more often it is measured. This pressure loss is usually determined in cold 
flow tests (with an inert fluid instead of the propellant and without combustion), 
and then the measured value is corrected for the actual propellant (different phys- 
ical properties) and the hot firing conditions; a higher temperature will change 
propellant densities or viscosities and in some designs it changes the cross section 
of cooling flow passages. 

Thrust Chamber Wall Loads and Stresses 

The analysis of loads and stresses is performed on all propulsion components 
during their engineering design. Its purpose is to assure the propulsion designer 



296 THRUST CHAMBERS 



and the flight vehicle user that (1) the components are strong enough to carry all 
the loads, so that they can fulfill their intended function; (2) potential failures have 
been identified, together with the possible remedies or redesigns; and (3) their 
masses have been reduced to a practical minimum. In this section we concentrate 
on the loads and stresses in the walls of thrust chambers, where high heat fluxes 
and large thermal stresses complicate the stress analysis. Some of the information 
on safety factors and stress analysis apply also to all propulsion systems, including 
solid propellant motors and electric propulsion. 

The safety factors (really margins for ignorance) are very small in rocket 
propulsion when compared to commercial machinery, where these factors can 
be 2 to 6 times larger. Several load conditions are considered for each rocket 
component; they are: 

1 . Maximum expected working load is the largest likely operating load under 
all likely operating conditions or transients. Examples are operating at a 
slightly higher chamber pressure than nominal as set by tolerances in design 
or fabrication (an example is the tolerance in setting the tank pressure 
regulator) or the likely transient overpressure from ignition shock. 

2. The design limit load is typically 1.20 times the maximum expected work- 
ing load, to provide a margin. If the variation in material composition, 
material properties, the uncertainties in the method of stress analysis, or 
predicted loads are significant, a larger factor may be selected. 

3. The damaging load can be based on the yield load or the ultimate load 
or the endurance limit load, whichever gives the lowest value. The yield 
load causes a permanent set or deformation, and it is typically set as 1.10 
times the design limit load. The endurance limit may be set by fatigue or 
creep considerations (such as in pulsing). The ultimate load induces a stress 
equal to the ultimate strength of the material, where significant elongation 
and area reduction can lead to failure. Typically, it is set as 1.50 times the 
design limit load. 

4. The proof test load is applied to engines or their components during devel- 
opment and manufacturing inspection. It is often equal to the design limit 
load, provided this load condition can be simulated in a laboratory. Thrust 
chambers and other components, whose high thermal stresses are difficult 
to simulate, use actual hot firing tests to obtain this proof, often with loads 
that approach the design limit load (e.g., higher than nominal chamber 
pressure or a mixture ratio that results in hotter gas). 

The walls of all thrust chambers are subjected to radial and axial loads from 
the chamber pressure, flight accelerations (axial and transverse), vibration, and 
thermal stresses. They also have to withstand a momentary ignition pressure 
surge or shock, often due to excessive propellant accumulation in the chamber. 
This surge can exceed the nominal chamber pressure. In addition, the chamber 
walls have to transmit thrust loads as well as forces and in some applications 
also moments, imposed by thrust vector control devices described in Chapter 18. 



8.2. COMBUSTION CHAMBER AND NOZZLE 297 



Walls also have to survive a “thermal shock,” namely, the initial thermal stresses 
at rapid starting. When walls are cold or at ambient temperature, they experience 
higher gas heating rates than after the walls have been heated. These loads are 
different for almost every design, and each unit has to be considered individually 
in determining the wall strengths. 

A heat transfer analysis is usually done only for the most critical wall regions, 
such as at and near the nozzle throat, at a critical location in the chamber, and 
sometimes at the nozzle exit. The thermal stresses induced by the temperature 
difference across the wall are often the most severe stresses and a change in 
heat transfer or wall temperature distribution will affect the stresses in the wall. 
Specific failure criteria (wall temperature limit, reaching yield stress, or maximum 
coolant temperature, etc.) have to be established for these analyses. 

The temperature differential across a chamber wall introduces a compressive 
stress on the inside and a tensile stress on the outside of the inner wall; the stress 
s can be calculated for simple cylindrical chamber walls that are thin in relation 
to their radius as 



s =2kE /S.T/{\-v) (8-12) 

where X is the coefficient of thermal expansion of the wall material, E the 
modulus of elasticity of the wall material, AT the temperature drop across the 
wall, and v the Poisson ratio of the wall material. Temperature stresses frequently 
exceed the yield point. The materials experience a change in the yield strength and 
the modulus of elasticity with temperature. The preceding equation is applicable 
only to elastic deformations. This yielding of rocket thrust chamber wall materials 
can be observed by the small and gradual contraction of the throat diameter after 
each operation (perhaps 0.05% reduction in throat diameter after each firing) and 
the formation of progressive cracks of the inside wall surface of the chamber and 
throat after successive runs. These phenomena limit the useful life and number 
of starts or temperature cycles of a thrust chamber (see Refs. 8-5 and 8-6). 

In selecting a working stress for the wall material of a thrust chamber, the 
variation of strength with temperature and the temperature stresses over the wall 
thickness have to be considered. The temperature drop across the inner wall is 
typically between 50 and 550 K, and an average temperature is sometimes used 
for estimating the material properties. The most severe thermal stresses can occur 
during the start, when the hot gases cause thermal shock to the hardware. These 
transient thermal gradients cause severe thermal strain and local yielding. 

A picture of a typical steady-state stress distribution caused by pressure loads 
and thermal gradients is shown in Fig. 8-11. Here the inner wall is subjected to a 
compressive pressure differential caused by a high liquid pressure in the cooling 
jacket and a relatively large temperature gradient. In a large rocket chamber, 
such as is used in the Redstone missile or the German V-2, the wall thickness 
of the nozzle steel way may be 7 mm and the temperature differential across it 
may readily exceed several hundred degrees. This temperature gradient causes 
the hot inner wall surface to expand more than the wall surface on the coolant 



298 THRUST CHAMBERS 



Gas film 




Liquid 

coolant 



Liquid film 



Typical temperature 
distribution 

Neutral axis (zero stress) 

1. Stress due to thermal expansion 
gradient across wall only 

- 2. Stress due to pressure differential 
across wall only 

3. Resultant stress (sum of curves 1 
and 2) with no yielding and constant 
modulus of elasticity 

■•4. Actual stress in wall with yielding 
at hot gas side and changing 
modulus of elasticity 

5. Yield stress distribution across 
wall (varies with temperature) 



I^Wall thickness— 



Distance from thrust chamber axis 



FIGURE 8-11. Typical stresses in a thrust chamber inner wall. 



side and imposes a high compressive thermal stress on the inside surface and 
a high tensile thermal stress on the coolant side. In these thick walls the stress 
induced by the pressure load is usually small compared to the thermal stress. 
The resultant stress distribution in thick inner walls (shown shaded in the sample 
stress diagram of Fig. 8-11) indicates that the stress in the third of the wall 
adjacent to the hot gases has exceeded the yield point. Because the modulus of 
elasticity and the yield point diminish with temperature, the stress distribution is 
not linear over the yielded portion of the wall. In effect, this inner portion acts 
as a heat shield for the outer portion which carries the load. 

Because of the differential expansion between the hot inner shell and the 
relatively cold outer shell, it is necessary to provide for axial expansion joints to 
prevent severe temperature stresses. This is particularly critical in larger double- 
walled thrust chambers. The German V-2 thrust chamber expanded over 5 mm 
in an axial and 4 mm in a radial direction. 

Tubes for tubular wall thrust chambers are subjected to several different 
stress conditions. Only that portion of an individual cooling tube exposed to 



8.2. COMBUSTION CHAMBER AND NOZZLE 299 



hot chamber gases experiences high thermal stresses and deformation as shown 
in Fig. 8-15. The tubes have to hold the internal coolant pressure, absorb the 
thermal stresses, and contain the gas pressure in the chamber. The hottest tem- 
perature occurs in the center of the outer surface of that portion of each tube 
exposed to hot gas. The thermal stresses are relatively low, since the temperature 
gradient is small; copper has a high conductivity and the walls are relatively thin 
(0.5 to 2 mm). The coolant pressure-induced load on the tubes is relatively high, 
particularly if the thrust chamber operates at high chamber pressure. The inter- 
nal coolant pressure tends to separate the tubes. The gas pressure loads in the 
chamber are usually taken by reinforcing bands which are put over the outside 
of the tubular jacket assembly (see Figs. 8-1 and 8-9). The joints between the 
tubes have to be gas tight and this can be accomplished by soldering, welding, 
or brazing. 

When a high-area-ratio nozzle is operated at high ambient pressure (sea level 
or low altitude), the lower part of the nozzle structure experiences a compression 
because the pressure in the nozzle near the exit is actually below atmospheric 
value. Therefore, high-area-ratio nozzles usually have stiffening rings on the 
outside of the nozzle near the exit to maintain a circular shape and thus prevent 
buckling, flutter, or thrust misalignment. 



Aerospike Thrust Chamber 

A separate category comprises thrust chambers using a center body, such as a 
plug nozzle or aerospike nozzle. They have more surface to cool than ordinary 
thrust chambers. A circular aerospike thruster is described in Chapter 3 and 
shown schematically in Figs. 3-12 and 8-12 and described in Ref. 8-7. Here the 
diameter of the exhaust flow plume increases with altitude. Each individual small 
(regeneratively cooled) thrust chamber or cell has its own cylindrical combustion 
chamber, a circular small nozzle throat, but a rectangular nozzle exit of low area 
ratio. The gas from these rectangular nozzle exits is further expanded (and thus 
reaches a higher exhaust velocity) along the contour of the spike or ramp. The flat 
surface at the bottom or base is porous or full of small holes and a low-pressure 
gas flows through these openings. This causes a back pressure on the flat base 
surface. This flow can be the exhaust gas from the turbopumps of an engine 
with a gas generator cycle; it is typically 1 or 2% of the total flow. The gas 
region below this base is enclosed by gas flows from the ramps. A linear version 
of a truncated (shortened) aerospike engine with two opposing ramps has been 
developed using LOX/LH 2 propellants at a thrust level of about 200,000 Ibf and 
the ground tests were satisfactory. It has two fuel-cooled side plates. Except for 
a flight of a very small, inefficient, truncated experimental aerospike, none of 
these liquid propellant engines have flown. 

The thrust of this aerospike thrust chamber consists of (1) the axial component 
thrusts of each of the little chamber modules or cells, (2) the integral of the 
pressures acting on the ramps over the axially projected area normal to the axis 
of the ramps, and (3) the pressure acting over the base area. 



300 THRUST CHAMBERS 



Propellant 
enters through 




FIGURE 8-12. Pressure profile and flow pattern along the ramp of an aerospike nozzle. 



The local gas pressures on the ramps are influenced by shock wave phenomena 
and change pressure profiles with altitude. Figure 8-12 shows a typical pressure 
distribution on a typical ramp surface and the flow patterns for low and high 
altitude. The hot gas flows coming out of the cell nozzles are turned into a nearly 
axial direction by multiple expansion waves (shown as dashed lines), which 
cause a reduction in pressure. At lower altitudes the turning causes compression 
shock waves (represented as solid lines), which causes local pressures to rise. 
The compression waves are reflected off the boundary between the hot gas jet 
and the ambient airstream, creating further expansion waves. At high altitude 
there are no compression waves emanating from the ramp and the expanding 
diverging flow exerts a decreasing pressure on the ramp area and behaves very 
similarly to the diverging nozzle flow in a bell-shaped conventional nozzle exit 
section. The contact locations, where the compression waves touch the ramp 
surface, have higher local heat transfer than other areas on the ramp surface; these 



8.3. LOW-THRUST ROCKETTHRUST CHAMBERS OR THRUSTERS 301 



locations move downstream as the vehicle gains altitude. The wave patterns and 
the pressure distribution on the spike or ramp surface can be determined from 
computerized fluid dynamics programs or a method of characteristics program. 

The advantages claimed for an aerospike engine are these: (1) Compared to 
rocket engines with conventional axisymmetric nozzles, the truncated aerospike 
is very short (about one third the length), and this also reduces the vehicle length 
and vehicle structure mass; (2) it has altitude compensation and thus operates at 
optimum nozzle expansion and highest possible exhaust velocity at every altitude; 
and (3) differential throttling of certain sets of individual thruster modules or cells 
allows pitch and yaw control, and with two engines it also allows roll control of 
the vehicle during powered flight, as explained in Chapter 18. There is no gimbal 
joint, no movement of the nozzle, no actuators, and no actuator power supply or 
strong structural locations for actuator side loads. The disadvantages include the 
lack of proven flight experience, proven reliability and performance validation, 
and a larger- than-usual surface area subject to high heat transfer. There are some 
small performance losses as the supersonic gas flow goes from a round nozzle 
throat of each cell into a rectangular nozzle exit of the cell, also in turning the 
flow on the ramp, and in combining supersonic flow from adjacent cell nozzles. 
The average total loss with liquid propellants is believed to be about 1% of 
the specific impulse. For combustion gases, which contain small particulates 
(e.g., gases from solid composite propellant explained in Chapter 13), the losses 
will be higher — up to 7% — depending on the percentage of particulates (which 
are tiny solid particles and/or small liquid droplets). Due to the inertia of the 
particulates, they can only be turned through a small angle in a high-speed gas 
flow and they will hit the nozzle walls. The mass flow distribution will be uneven 
with the heavy particulates mostly in the gas layers next to the concave curved 
nozzle walls. Ref. 8-8 gives an example, which did not recognize the loss from 
particulates. 



8.3. LOW-THRUST ROCKET THRUST CHAMBERS OR THRUSTERS 

Spacecraft, certain tactical missiles, missile defense vehicles, and upper stages of 
ballistic missiles often use special, multiple thrusters in their small, liquid pro- 
pellant rocket engines. They generally have thrust levels between about 0.5 and 
10,000 N or 0. 1 to 2200 Ibf, depending on vehicle size and mission. As mentioned 
in Section 4.5, they are used for trajectory corrections, attitude control, docking, 
terminal velocity control in spacecraft or ballistic missiles, divert or side move- 
ment, propellant settling, and other functions. Most operate with multiple restarts 
for relatively short durations during a major part of their duty cycle. As mentioned 
in Section 6.7, they can be classified into hot gas thrusters (high-performance 
bipropellant with combustion temperatures above 2600 K and vacuum of 230 
to 325 sec), warm gas thrusters such as monopropellant hydrazine (temperatures 
between 500 and 1600 K and /, of 180 to 245 sec), and cold gas thrusters such 
as high-pressure stored nitrogen (200 to 320 K) with low specific impules (40 to 
120 sec). 



302 THRUST CHAMBERS 



A typical small thruster for bipropellant is shown in Fig. 8-13 and for 
hydrazine monopropellant in Fig. 8-14. For attitude control angular motions 
these thrust chambers are usually arranged in pairs as explained in Section 4.5. 
The same control signal activates the valves for both units of such a pair. All 
these small space rocket systems use a pressurized feed system, some with pos- 
itive expulsion provisions, as explained in Section 6.2. The vehicle mission and 
the automatic control system of the vehicle often require irregular and frequent 
pulses to be applied by pairs of attitude control thrust chambers, which often 
operate for very short periods (as low as 0.01 to 0.02 sec). This type of frequent 
and short-duration thrust application is also known as pulsed thrust operation. 
For translation maneuvers a single thruster can be fired (often in a pulsing mode) 
and the thrust axis usually goes through the center of gravity of the vehicle. The 
resulting acceleration will depend on the thrust and the location of the thruster 
on the vehicle; it can be axial or at an angle to the flight velocity vector. 

There is a performance degradation with decreasing pulse duration for all three 
types of thrusters, because propellants are used inefficiently during the buildup 
of thrust and the decay of thrust, when they operate below full chamber pressure 
and the nozzle expansion characteristics are not optimum. The specific impulse 



Niobium chamber with disilicide coating (0.003 in. thick) 




2 Propellant valves 

FIGURE 8-13. This radiation-cooled, insulated vernier thruster is one of several used 
on the reaction control system of the Space Shuttle vehicle for orbit stabilization and 
orientation, rendezvous or docking maneuvers, station keeping, deorbit, or entry. The 
nozzle is cut off at an angle to fit the contour of the vehicle. Performance data are given 
in Table 6-5. Operation can be in a pulse mode (firing durations between 0.08 and 
0.32 sec with minimum offtime of 0.08 sec) or a steady-state mode (0.32 to 125 sec). 
Demonstrated life is 23 hr of operation and more than 300,000 starts. (Courtesy of Aerojet 
Propulsion Company and NASA.) 




8.3. LOW-THRUST ROCKETTHRUST CHAMBERS OR THRUSTERS 303 





Shower head 

FIGURE 8-14. Typical hydrazine monopropellant small thrust chamber with catalyst 
bed, showing different methods of injection. 



really suffers when the pulse duration becomes very short. In Section 3.5 the 
actual specific impulse of a rocket operating at a steady state was given at about 
92% of theoretical specific impulse. With very short pulses (0.01 sec) this can 
be lower than 50%, and with pulses of 0.10 sec it can be around 75 to 88%. 
Also, the reproducibility of the total impulse delivered in a short pulse is not 
as high after prolonged use. A preheated monopropellant catalyst bed will allow 
performance improvement in the pressure rise and in short pulses. 

One way to minimize the impulse variations in short pulses and to maximize 
the effective actual specific impulse is to minimize the liquid propellant passage 
volume between the control value and the combustion chamber. The propellant 
flow control valves for pulsing attitude control thrust chambers are therefore 
often designed as an integral part of the thrust chamber-injector assembly, as 
shown in Fig. 8-13. Special electrically actuated leakproof, fast-acting valves 
with response times ranging from 2 to 25 msec for both the opening and closing 



304 THRUST CHAMBERS 



operation are used. Valves must operate reliably with predictable characteristics 
for perhaps 40,000 to 80,000 starts. This in turn often requires endurance proof 
tests of 400,000 to 800,000 cycles. 

Throttling or reducing the thrust is equivalent to reducing the total impulse 
over the operating duration. This can be achieved by varying (1) the time between 
pulses (less total cumulative impulse per unit time) or (2) by limiting the total 
number of pulses. 

Liquid storable bipropellants such as N204-monomethylhydrazine are used 
when high performance is mandatory. Some have used ablative materials for 
thrust chamber construction, as in the Gemini command module. The Space 
Shuttle small thrusters use radiation cooling with refractory metals, as shown 
in Fig. 8-13. A radiation-cooled thruster is shown later in Fig. 8-16. Carbon 
materials made of woven strong carbon fibers in a matrix of carbon have also 
been used in radiation-cooled bipropellant thrusters. 

Hydrazine monopropellant thrusters are used when system simplicity is impor- 
tant and moderate performance is acceptable. They have a nontoxic, clear, clean 
exhaust plume. Virtually all hydrazine monopropellant attitude control rockets 
use finely dispersed iridium or cobalt deposited on porous ceramic (aluminum 
oxide) substrate pellets 1.5 to 3 mm in diameter as a catalyst. Figure 8-14 shows 
a typical design of the catalyst pellet bed in an attitude control thruster designed 
for pulse and steady-state operation meeting a specific duty cycle. Each injection 
passage is covered with a cylindrical screen section which extends into a part of 
the catalyst bed and distributes the hydrazine propellant. Figure 8-14 also shows 
other successful types of hydrazine injector. Several arrangements of catalyst 
beds are employed; some have spring-loading to keep the pellets firmly packed. 
Hydrazine monopropellant thrust units range in size from 0.2 to 2500 N of thrust; 
the scaling procedure is empirical and each size and design requires testing and 
retesting. The amount of ammonia decomposition, shown in Fig. 7-3, can be 
controlled by the design of the catalyst bed and its decomposition chamber. 

Mechanical, thermal, and chemical problems arise in designing a catalyst bed 
for igniting hydrazine, the more important of which are catalytic attrition and 
catalyst poisoning. Catalytic attrition or physical loss of catalyst material stems 
from motion and abrasion of the pellets, with loss of very fine particles. Crushing 
of pellets can occur because of thermal expansion and momentary overpressure 
spikes. As explained in Chapter 7, the catalyst activity can also decline because of 
poisoning by trace quantities of contaminants present in commercial hydrazine, 
such as aniline, MMH, UDMH, sulfur, zinc, sodium, or iron. Some of these 
contaminants come with the hydrazine and some are added by the materials 
used in the tanks, and propellant plumbing in the spacecraft. The high-purity 
grade hydrazine has less than 0.003% aniline and less than 0.005% carbonaceous 
material; it does not contaminate catalysts. Catalyst degradation, regardless of 
cause, produces ignition delays, overpressures, and pressure spikes, decreases 
the specific impulse, and decreases the impulse bit per pulse in attitude control 
engines. 



8.4. MATERIALS AND FABRICATION 305 



Figure 17-4 shows a combination of chemical monopropellant and electrical 
propulsion. Electrical postheating of the reaction gases from catalysis allows an 
increase of the altitude specific impulse from 240 sec to about 290 or 300 sec. 
A number of these combination auxiliary thrusters have successfully flown on a 
variety of satellite applications and are particularly suitable for spacecraft where 
electrical power is available and extensive short-duration pulsing is needed. 

Cold gas thrusters use stored high-pressure inert gas as the propellant; they 
and their performance were mentioned in Section 7.6 and their propellants and 
specific impulses are listed in Table 7-3. They are simple, low cost, used with 
pressurized feed systems, used for pulsing operations, and for low thrust and low 
total impulse. They can use aluminum or plastics for thrusters, valves, and piping. 
The early versions of the Pegasus air-launched launch vehicle used them for roll 
control. The advantages of cold gas systems are: (a) they are very reliable and 
have been proven in space flights lasting more than 10 years; (b) the system is 
simple and relatively inexpensive; (c) the ingredients are nontoxic; (d) no deposit 
or contamination on sensitive spacecraft surfaces, such as mirrors; (e) they are 
safe; and (f) capable of random pulsing. The disadvantages are: (a) engines are 
relatively heavy with poor propellant mass fractions (0.02 to 0.19); (b) the specific 
impulses and vehicle velocity increments are low, when compared to mono- or 
bipropellant systems; and (c) relatively large volumes. 



8.4. MATERIALS AND FABRICATION 

The choice of the material for the inner chamber wall in the chamber and the 
nozzle throat region, which are the critical locations, is influenced by the hot 
gases resulting from the burning of propellants, the maximum wall temperature, 
the heat transfer, and the duty cycle. A large variety of material has been used. 
Table 8-3 lists typical selected materials for several applications, thrust sizes, 
and propellants. For high-performance, high heat transfer, regeneratively cooled 
thrust chambers a material with high thermal conductivity and a thin-wall design 
will reduce the thermal stresses. Copper is an excellent conductor and it will not 
really oxidize in fuel-rich noncorrosive gas mixtures. It is often used with oxygen 
and hydrogen below a mixture ratio of 6.0. The inner walls are therefore usually 
made of a copper alloy (with small additions of zirconium, silver, or silicon), 
which has a conductivity not quite as good as pure (oxygen-free) copper but has 
improved high-temperature strength. 

Figure 8-15 shows a partial section through five configurations of cooling 
jackets. All involve curved or doubly curved components and precision fits for 
good fastening or joining. The top configuration has an intermediate thin corru- 
gated sheet of metal and has been used extensively in Russia and its predecessor 
the Soviet Union. It is soldered together under external pressure in a furnace. It 
is used in those locations on a thrust chamber, where the heat transfer is mild 
or of intermediate intensity. The milled slot design can usually accept the high- 
est heat transfer intensities. One fabrication technique is to machine (usually on 



306 THRUST CHAMBERS 



a milling machine) nearly rectangular grooves or cooling channels of varying 
width and depth into the inner surface of a relatively thick contoured inner wall 
using a high conductivity material such as a copper alloy. The grooves are then 
filled with wax and by an electrolytic plating technique build an outer wall of a 
suitable metal (such as nickel) so as to enclose the coolant channels. The wax 
is then melted out. The third design shows a tubular construction; it has been 
used extensively in the United States for larger thrust chambers with intermediate 
to high heat transfer. Figure 8-9 shows that individual tubes are shaped to the 
contour of the thrust chamber including its nozzle; the round tube shape is altered 
into nearly rectangular cross sections. The formed and brazed tubes are brazed 
or soldered together in a furnace and the outer structural shell or outer bands are 
also brazed to the assembly. The corrugated outer wall concept is perhaps the 
simplest and often the lightest cooling jacket configuration. It has been used in 
locations where heat transfer is modest. The bottom shell represents a welded 
construction, made of stainless steel. It has a higher allowable maximum wall 
temperature than a brazed construction. It has been used in nozzles designed and 
built in Europe. 



Outer wall 

Intermediate corrugated sheet 
Inner wall 




(a) intermediate corrugated brazed 
sheet forms the cooling channels. 




Outer shell bonded to tops 
of channel walls 



Milled cooling channel 
Inner wall 



(b) Milled channel design. 




Outer struaural bands or shell 
Tubes formed 
Cooling passage 



(c) Tubular cooling Jacket. 



Corrugated outer wall 

Cooling channel 

inner wall 




(d) Corrugated outer wail. 




(e) Welded formed "Z"-shaped channels. 

FIGURE 8-15. Simplified sketches of sections through a portion of the cooling jacket 
of several different cooling schemes in regeneratively cooled thrust chambers. (Used with 
permission from Ref. 8-1.) 



8.4. MATERIALS AND FABRICATION 307 



TABLE 8-3. Typical Materials Used in Several Common Liquid Propellant Thrust 
Chambers 









Cooling 




Application 


Propellant 


Components 


Method 


Typical Materials 


Bipropellant TC, 


Oxygen — 


C, N, E 


E 


Copper alloy 


cooled, high 
pressure (booster or 
upper stage) 


hydrogen 


I 


E 


Transpiration cooled 










porous stainless 
steel face. Structure 
is stainless steel 






Alternate E 


R 


Carbon fiber in a 










carbon matrix, or 
niobium 






Alternate E 


T 


Steel shell with 










ablative inner liner 


Bipropellant TC 


Oxygen- 


C, N, E, I 


E 


Stainless steel with 


cooled, high 


hycrocarbon 






tubes or milled 


pressure (booster or 


or storable 






slots 


upper stage) 


propellant" 


Alternate E 


R 


Carbon fiber in a 










carbon matrix, or 
niobium 






Alternate E 


T 


Steel shell with 










ablative inner liner 


Experimental TC 


All types 


C, N, E 


U 


Low carbon steel 


(very limited 
duration — only a 
few seconds) 










Small bipropellant TC 


All types 


C, N, E 


R 


Carbon fiber in 










carbon matrix, 
rhenium, niobium 








T 


Steel shell with 










ablative inner 
linear 






I 


E 


Stainless steels. 










titanium 


Small monopropellant 


Hydrazine 


C, N, E, 


R 


Inconel, alloy steels 


TC 




I 


E 


Stainless steel 


Cold gas TC 


Compressed 


C, N, E, I 


U 


Aluminum, steel, or 




air, nitrogen 






plastic 



“HNOs or N2O4 oxidizer with N2H4, MMH, or UDMH as fuels (see Chapter 7 ). TC = thrust 
chamber, C = chamber wall, N = nozzle converging section walls, diverging section walls, and 
throat region walls, E = walls at exit region of diverging section of nozzle, I = injector face, F = fuel 
cooled (regenerative), R = radiation cooled, U = uncooled, T = transient heat transfer or heat sink 
method (ablative material). 



308 THRUST CHAMBERS 



The depth and width of the cooling channels vary with the chamber pro- 
hle location and the local thrust chamber diameter. At the nozzle throat region 
the heat transfer is the highest; therefore it is the location of the highest cool- 
ing velocity; therefore the total cooling passage area at the throat region is the 
smallest. Often two or sometimes three different cooling jacket constructions are 
designed into the same thrust chamber. Typically, the milled groove configuration 
is used for the nozzle throat region and other configurations are then used for the 
chamber and the nozzle exit regions. 

The failure modes often are bulging on the hot wall on the gas side and the 
opening up of cracks. During hot firing, fhe strain at the hot surface can exceed 
the local yield point, thus giving it a local permanent compressive deformation. 
With the cooldown after operation and with each successive firing, some addi- 
tional yielding and further plastic deformation will occur until cracks form. With 
successive firings the cracks can become deep enough for a leak of cooling pro- 
pellant into the chamber or nozzle and the thrust chamber will then usually fail. 
The useful life of a metal thrust chamber is the maximum number of firings 
(and sometimes also the cumulative bring duration) without such a failure. The 
prediction of wall failures is not simple and Refs. 8-5 and 8-6 explain this in 
more detail. Useful life can also be limited by the storage life of soft compo- 
nents (0-rings, gaskets, valve stem lubricant) and, for small thrusters with many 
pulses, also the fatigue of valve seats. Therefore, there is a maximum limit on 
the number of brings that such a thrust chamber can withstand safely, and this 
limits its useful life. 

For radiation cooling, several different carbon materials have worked in a 
reducing, fuel-rich rocket combustion atmosphere. At leaner gas mixtures the 
carbon can oxidize at the elevated temperatures when they glow red or white. 
They can be used at wall temperatures up to perhaps 3300 K or 6000°R. Carbon 
materials and ablative materials are used extensively in solid propellant rocket 
motors and are discussed further in Chapter 15. 

For some small radiation-cooled bipropellant thrusters with storable propel- 
lants, such as those of the reaction control thrusters on the Space Shuttle orbiter, 
the hot walls are made of niobium coated with disilicide (up to 1120 K or 
2050°R). To prevent damage, a fuel-rich mixture or blm cooling is often used. 
For small thrusters rhenium walls protected by iridium coatings (oxidation resis- 
tant) have come into use more recently and have been used up to about 2300 K or 
4100°R (see Refs. 8-1 and 8-9). Other high-temperature materials, such as tung- 
sten, molybdenum, alumina, or tantalum, have been tried but have had problems 
in manufacture, cracking, hydrogen embrittlement, or excessive oxidation. 

A small radiation-cooled bipropellant thruster is shown in Fig. 8-16. It uses 
three different nozzle and chamber materials. This thruster’s injector has extra 
fuel injection holes (not shown in Fig. 8-16) to provide blm cooling to keep wall 
temperatures below their failure limits. High temperature copper-nickel alloys or 
stainless steels are used for the radiation cooled nozzle-and chamber walls of 
thrusters operating with hydrazine monopropellant. See Fig. 8-14. 



8.4. MATERIALS AND FABRICATION 309 




FIGURE 8-16. Radiation-cooled reaction control thruster R-4D-15 uses nitrogen tetrox- 
ide and monomethylhydrazine propellants. The large nozzle area ratio allows good vacuum 
performance. It has three different nozzle materials, each with a lower allowable temper- 
ature (Re 4000°F; Nh 3500°F; Ti 1300°F). Courtesy of Aerojet Propulsion Company. 



Until recently it has not been possible to make large pieces of carbon-carbon 
material. This was one of the reasons why large nozzle sections and integral 
nozzle exit cone pieces in solid motors were made from carbon phenolic cloth 
lay-ups. Progress in manufacturing equipment and technology has now made it 
possible to build and fly larger pieces made of carbon fiber in a carbon matrix. A 
three-piece extendible carbon nozzle exit cone of 2.3 m (84 in.) diameter and 2.3 
to 3 mm thickness has flown on an upper-stage engine. This thrust chamber with 
its movable nozzle extension is shown in Fig. 8-17, its parameters are listed in 
Table 8-1, its testing is reported in Ref. 8-4, and it has flown successfully many 
times. 

The material properties have to be evaluated under all likely operating condi- 
tions, loads, start conditions, pressure variations, and the like before a material 
can be selected for a specific thrust chamber application. This evaluation includes 
physical properties, such as tensile and compressive strengths, yield strength, frac- 
ture toughness, modulus of elasticity (for determining deflections under load), 
thermal conductivity (a high value is best for steady-state heat transfer), coef- 
ficient of thermal expansion (some large thrust chambers grow by 3 to 10 mm 
when they become hot, and that can cause problems with their piping connec- 
tions or structural supports), specific heat (capacity to absorb thermal energy), 
reflectivity (for radiation), or density (ablatives require more volume than steel). 



310 THRUST CHAMBERS 



Nozzle extension device (motors) 



Gimbal mount 




Controller 

Regeneratively 
cooled chamber 
and nozzle 

Propellant 

cooldown 

lines 

Movable 
nozzle extension 



(a) Half section of nozzle extension in stowed position 




FIGURE 8-17. The RL-lOB-2 rocket engine has an extendible nozzle cone or skirt, 
which is placed around the engine during the ascent of the Delta 111 and IV launch 
vehicles. This extension is lowered into position by electromechanical devices after the 
launch vehicle has been separated from the upper stage at high altitude and before firing. 
(Courtesy of Pratt & Whitney Rocketdyne, of United Technologies, Inc.) 



8.5. HEAT TRANSFER ANALYSIS 311 



These properties change with temperature (they are different when they are hot) 
and sometimes they change with little changes in materials composition. The 
temperature where a material loses perhaps 60 to 75% of its room temperature 
strength is often selected as the maximum allowable wall temperature, well below 
its melting point. Since a listing of all the key properties of a single material 
requires many pages, it is not possible to list them here, but they are usually 
available from manufacturers and other sources. Other important properties are 
erosion resistance, little or no chemical reactions with the propellants or the hot 
gases, reproducible decomposition or vaporization of ablative materials, ease and 
low cost of fabrication (welding, cutting, forming, etc.), the consistency of the 
composition (impurities) of different batches of each material (metals, organics, 
seals, insulators, lubricants, cleaning fluids), and ready availability and low cost 
of each material. 



8.5. HEAT TRANSFER ANALYSIS 

In actual rocket development not only is the heat transfer analyzed but the rocket 
units are almost always tested to assure that heat is transferred satisfactorily under 
all operating and emergency conditions. Heat transfer calculations are useful to 
guide the design, testing, and failure investigations. Those rocket combustion 
devices that are regeneratively cooled or radiation cooled can reach thermal equi- 
librium and the steady-state heat transfer relationships will apply. Transient heat 
transfer conditions apply not only during thrust buildup (starting) and shutdown 
of all rocket propulsion systems, but also with cooling techniques that never 
reach equilibriurm, such as with ablative materials. 

Sophisticated finite element analysis (FEA) programs of heat transfer have 
been available for at least 20 years, and several different FEA computer programs 
have been used for the analysis of thrust chamber steady-state and transient 
heat transfer, with different chamber geometries or different materials and with 
temperature-variant properties. A detailed description of this powerful analysis 
is beyond the scope of this book, but can be found in Refs. 8-10, 8-11, and 
8-12. Major rocket propulsion organizations have developed their own versions 
of suitable computer programs for solving their heat transfer problems. This 
section gives the basic relationships that are the foundation for FEA programs. 
They are intended to give some understanding of the phenomena and underlying 
principles. 



General Steady-State Heat T ransfer Relations 

For heat transfer conduction the following general relation applies: 

Q dT AT 



(8-13) 



312 THRUST CHAMBERS 



where Q is the heat transferred across a surface area A, dT/dL the temperature 
gradient with respect to thickness at the surface A, and k the thermal con- 
ductivity expressed as the amount of heat transferred per unit time through a 
unit area of surface for 1° temperature difference over a unit wall thickness. The 
negative sign indicates that temperature decreases as thickness increases. 

The steady-state heat transfer through the chamber wall of a liqud-cooled 
rocket chamber can be treated as a series-type, steady-state heat transfer problem 
with a large temperature gradient across the gaseous film on the inside of the 
chamber wall, a temperature drop across the wall, and, in cases of cooled cham- 
bers, a third temperature drop across the film of the moving cooling fluid. It is a 
combination of convection at the boundaries of the flowing fluids and conduction 
through the chamber walls. It is shown schematically in Fig. 8-18. 

The general steady-state heat transfer equations for regeneratively cooled thrust 
chambers can be expressed as follows: 



h{Tg-Ti) = Q/A 


(8-14) 


Tg-Ti 

l/hg + t„ 1 K -F l/hi 


(8-15) 




(8-16) 


{K/t„)(T„g - T„i) 


(8-17) 


hi{T„i — Ti) 


(8-18) 



where q is heat transferred per unit area per unit time, Tg the absolute chamber 
gas temperature, T/ the absolute coolant liquid temperature, T^,i the absolute wall 
temperature on the liquid side of the wall, T^g the absolute wall temperature on 




FIGURE 8-18. Temperature gradients in cooled rocket thrust chamber. The listed tem- 
peratures are typical. 



8.5. HEAT TRANSFER ANALYSIS 313 



the gas side of the wall, h the overall film coefficient, hg the gas film coefficient, 
hi the coolant liquid film coefficient, the thickness of the chamber wall, and 
K the conductivity of the wall material. The strength and thermal properties of 
materials are functions of temperature. Any consistent set of units can be used 
in these equations. These simple relations assume that the heat flow is radial. 
The simple quasi-one-dimensional theory also often assumes that the thermal 
conductivity and the film coefficients are at average values and not functions 
of temperature or pressure. A two- or three-dimensional finite element model 
would also need to be used to analyze the heat transfer in the axial directions, 
which usually occurs in the nozzle throat wall regions; some of the heat from 
the hot nozzle throat is transferred to wall regions upstream and downstream of 
the throat. 

Because the film coefficients, the gas and liquid coolant temperatures, the wall 
thickness, and the surface areas usually vary with the axial distance L within 
a combustion chamber (assuming axial heat transfer symmetry), the total heat 
transfer per unit time Q can be found by integrating the local heat transfer over 
the entire internal surface area of the chamber and the nozzle: 



Because both q and D are complicated functions of the thrust chamber length 
L, the equation usually has to be solved by dividing the rocket chamber into finite 
lengths. Assuming that q is given by Eqs. 8-13 to 8-19 and remains constant 
over the length of each element gives an approximate solution. 

The important quantities for controlling the heat transfer across a rocket cham- 
ber wall are the fluid film boundaries established by the combustion products on 
one side of the wall and the coolant flow on the other. The gas film coefficient 
largely determines the numerical value of the heat transfer rate, and the liquid 
film largely determines the value of the wall temperatures. The determination of 
the film coefficients in Eqs. 8-16 and 8-18 is difficult because of the complex 
geometries, the nonuniform velocity profile, the surface roughness, the boundary 
layer behavior, and the combustion oscillations. 

Conventional heat transfer theory is usually given in terms of several dimen- 
sionless parameters (Ref. 8-10, 8-11, and 8-12): 



where hg is the film coefficient, D the diameter of the chamber of the nozzle, 
V the calculated average local gas velocity, ic the conductivity of the gas, /u, the 
absolute gas viscosity, Cp the specific heat of the gas at constant pressure, and p 
the gas density. 

In Eq. 8-20 the quantity hgD /k is known as the Nusselt number, the quantity 
Dvp/ pL as the Reynolds number, and the quantity Cppt/K as the Prandtl number 




(8-19) 




( 8 - 20 ) 



314 THRUST CHAMBERS 



Pr. The gas film coefficient hg can be determined from Eq. 8-21: 




( 8 - 21 ) 



where pn is the local mass velocity, and the constant 0.026 is dimensionless. The 
boundary layer temperature gradient has effects on the various gas properties in 
rocket combustion, Conventional theoretical approaches, such as for Eq. 8-20 
or 8-21, are usually for heated circular tubes, relatively long tubes, steady-state 
flow, and they reach an equilibrium velocity profile. Heat flows into the tube from 
all sides (360°). However, in thrust chambers the heat flow to the coolant passage 
is only on one side of the cooling passage and the combustion phenomena are 
propellant specific. The chamber length is relatively short, and an equilibrium 
flow profile does not exist. The actual flow in the combustion chamber is highly 
turbulent. There are liquid droplets, which evaporate, and there is no equilibrium. 
For these reasons Eqs. 8-20 and 8-21 an approximations only. 

Equations where the coefficients have been validated by actual experimen- 
tal data are more reliable and they are used in design. Bartz (Ref. 8-13) has 
surveyed the agreement between theory and experiment and developed semiem- 
pirical correction factors: 



The subscript 0 refers to properties evaluated at the stagnation or combustion 
temperature; the subscript am refers to properties at the arithmetic mean temper- 
ature of the local free-stream static temperature and the wall temperatures; and 
p' is the free-stream value of the local gas density. Again, the empirical constant 
0.026 is dimensionless when compatible dimensions are used for the other terms. 
The gas velocity v is the local free-stream velocity corresponding to the density 
p'. Since density raised to the 0.8 power is roughly proportional to the pressure 
and the gas film coefficient is roughly proportional to the heat flux, it follows that 
the heat transfer rate increases approximately linearly with the chamber pressure. 
These semiempirical heat transfer equations have been modified and validated 
for common propellants, limited chamber pressure ranges, and specific injectors 
(see Ref. 8-13) and are often proprietary to specific design organizations. 

The temperature drop across the inner wall and the maximum temperature are 
reduced if the wall is thin and is made of material of high thermal conductiv- 
ity. The wall thickness is determined from strength considerations and thermal 
stresses, and some designs have as little as 0.025 in. thickness. The effect of 
changing the film coefficients is shown in Example 8-1. 

Surface roughness can have a large effect on the film coefficients and thus on 
the heat flux. Measurements have shown that the heat flow can be increased by 
a factor of up to 2 by surface roughness and to higher factors when designing 
turbulence-creating obstructions in the cooling channels. Major surface roughness 




( 8 - 22 ) 



8.5. HEAT TRANSFER ANALYSIS 315 



on the gas side will cause the gas locally to come close to stagnation temper- 
ature. However, surface roughness on the liquid coolant side of the wall will 
enhance turbulence and the absorption of heat by the coolant and reduce wall 
temperatures. 

Example 8-1. The effects of varying the film coefficients on the heat transfer and the 
wall temperatures are to be explored. The following data are given: 

Wall thickness 0.445 mm 

Wall material Low-carbon steel 

Average conductivity 43.24 W/m^- K/m 

Average gas temperature 3033 K or 2760°C 

Average liquid bulk temperature 311.1 K or 37.8°C 

Gas-film coefficient 147 W/m^-°C 

Liquid-film coefficient 205,900 W/m^-°C 



Vary hg (at constant hi), then vary hi (at constant hg), and then determine the changes in 
heat transfer rate and wall temperatures on the liquid and the gas side of the wall. 

SOLUTION. Use Eqs. 8-13 to 8-18 and solve for q, T„g, and The answers shown 
in Table 8-4 indicate that variations in the gas-film coefficient have a profound influence 
on the heat transfer rate but relatively little effect on the wall temperature. The exact 
opposite is true for variations in the liquid-film coefficient; here, changes in hi produce 
little change in q but a fairly substantial change in the wall temperature. 



TABLE 8-4. Change in Film Coefficient for Example 8-1 



Change in Film 
Coefficient (%) 


Change in 
Heat Transfer 

(%) 


Wall Temperature (K) 


Gas Film 


Liquid Film 


Gas Side, Ty„g 


Liquid Side, T^i 


50 


100 


50 


324.4 


321.1 


100 


100 


100 


337.2 


330.5 


200 


100 


198 


362.8 


349.4 


400 


100 


389 


415.6 


386.1 


100 


50 


99 


356.1 


349.4 


100 


25 


98 


393.3 


386.7 


100 


12.5 


95 


460.0 


397.8 


100 


6.25 


91 


596.7 


590.5 



Figure 8-19 shows heat flow directions, temperature distribution, and the 
locations for the maximum wall temperatures for a milled cooling jacket design. 
This design is represented by the third sketch of Fig. 8-15. The inner wall should 
be thin, so that the temperature difference across this wall is low and therefore 
the thermal stresses are also low. 



316 THRUST CHAMBERS 



Estimated 

temperature 

distribution 




Hottest areas 



Isotherms (solid lines) 

Dashed lines indicate 
direction of heat flux 



FIGURE 8-19. Results of a two-dimensional analysis of the heat transfer in two cooling 
channels of a milled slot cooling jacket. The outer wall and the upper parts of the channels 
are not shown. 



T ransient Heat T ransfer Analysis 

An uncooled (high melting point) metal thrust chamber is the simplest type to 
analyze because there is no chemical change. Thermal equilibrium is not reached. 
The uncooled walls act essentially as a heat sponge and absorb heat from the hot 
gases. With the aid of experimental data to determine some typical coefficients, 
it is possible in some cases to predict the transient heating of uncooled walls. 

Heat is transferred from the hot gases to the wall, and during operation a 
changing temperature gradient exists across the wall. The heat transferred from 
the hot wall to the surrounding atmosphere, and by conduction of metal parts to 
the structure, is negligibly small during this transient heating. Each local point 
within the wall has its temperature raised as the burning process is extended in 
time. After the completion of the rocket’s operation, the wall temperatures tend 
to equalize. A typical temperature -time -location history is given in Fig. 8-20. 
Here the horizontal line at T = 21°C denotes the initial equilibrium condition 
of the wall before the rocket operates; the various curves show the temperature 
profile across the wall at successive time intervals after initiation of combustion. 
The line at T = 357°C shows an equilibrium temperature of the wall a finite time 
after cutoff. 

The heat transferred across the hot surface of the wall (and distributed within 
the wall by conduction) must be less than the heat-absorbing capacity of the wall 
material below the critical temperature. If heat transfer to the outside atmosphere 
and axially within the metal wall is neglected, this can be expressed in a simplified 
form: 

g At = -KA{dT/dL)At = me AT (8-23) 

where Q is the heat per second transferred across an inner wall surface area A. 
Equation 8-14 shows that Q/A depends on the hot gas temperature, the wall 
temperature, and the gas film coefficient. The heat conductivity k depends on the 
material and its temperature; AT denotes the average wall temperature increment; 
dT/dL the temperature gradient of the heat flow near the hot wall surface in 
degrees per unit thickness; m the mass of a unit area of wall; c the average 
specific heat of the wall material; and At is the time increment. The chamber and 
nozzle walls can be divided into cylindrical or conical segments, and each wall 



8.5. HEAT TRANSFER ANALYSIS 317 




Wall position number 

Wall thickness, 12.7 mm H 

Heated Insulated 

surface surface 

FIGURE 8-20. Typical temperature distributions through a wall of an uncooled relatively 
thick metal thrust chamber as a function of heating time. 



segment in turn is divided into an arbitrary number of axisymmetric concentric 
layers, each of a finite thickness. At any given time the heat conducted from any 
one layer of the wall exceeds the heat conducted into the next outer layer by 
the amount of heat absorbed in raising the temperature of the particular layer. 
This iterative approach lends itself readily to two- or three-dimensional computer 
analysis, resulting in data similar to Fig. 8-20. It is usually sufficient to determine 
the heat transfer at the critical locations, such as in the nozzle throat region. 



318 THRUST CHAMBERS 



A more complex three-dimensional analysis can also be undertaken; here the 
wall geometry is often more complex than merely cylindrical, heat is conducted 
also in directions other than normal to the axis, temperature variable properties 
are used, boundary layer characteristics vary with time and location, and there 
may be more than one material layer in the wall. 

A number of mathematical simulations of transient heat transfer in ablative 
materials have been derived, many with limited success. This approach should 
include simulation for the pyrolysis, chemical decomposition, char depth, and 
out-gassing effects on film coefficient, and it requires good material property 
data. Most simulations require some experimental data. 

Steady-State Transfer to Liquids in Cooling Jacket 

The term regenerative cooling is used for rockets where one of the propellants is 
circulated through cooling passages around the thrust chamber prior to the injec- 
tion and burning of this propellant in the chamber. It is really forced convection 
heat transfer. The term regenerative is perhaps not altogether appropriate here, 
and it bears little relation to the meaning given to it in steam turbine practice. It 
is intended to convey the fact that the heat absorbed by the coolant propellant is 
not wasted but augments its initial temperature and raises its energy level before 
it passes through the injector. This increase in the internal energy of the liquid 
propellant can be calculated as a correction to the enthalpy of the propellant 
(see Chapter 5). However, the overall effect on rocket performance is usually 
very slight. With some propellants the specific impulse can be 1% larger if the 
propellants are preheated through a temperature differential of 100 to 200° C. 
In hydrogen-cooled thrust chambers and in small combustion chambers, where 
the wall-surface-to-chamber volume ratio is relatively large, the temperature rise 
in the regenerative coolant will be high, and the resulting increase in specific 
impulse is sometimes more than 1%. 

The behavior of the liquid film is critical for controlling the wall temperatures 
in forced convection cooling of rocket devices at high heat fluxes (see Table 8-4 
and Refs. 8-10, 8-14, and 8-15). At least four different types of film appear 
to exit, as can be interpreted from Fig. 8-21. Here the heat transfer rate per 
unit of wall surface q is shown as a function of the difference between the wall 
temperature on the liquid side T„i and the bulk temperature of the liquid T;. 

1. The normal forced convection region at low heat flux appears to have 
a liquid boundary layer of predictable characteristics. It is indicated by 
region A-B in Fig. 8-21. Here the wall temperature is usually below the 
boiling point of the liquid at the cooling jacket pressure. In steady-state 
heat transfer analysis the liquid-film coefficient can be approximated by 
the usual equation (see Refs. 8-10, 8-11, and 8-12): 




(8-24) 



8.5. HEAT TRANSFER ANALYSIS 



319 




FIGURE 8-21. Regimes in transferring heat from a hot wall to a flowing liquid. 



where m is the liquid fluid mass flow rate, c its average specific heat, 
A the cross-sectional cooling jacket flow area, D the equivalent diameter 
of the coolant passage cross section,* v the fluid velocity, p the coolant 
density, /r its absolute viscosity, and /c its conductivity. Many liquid-cooled 
rocket devices operate in this regime of heat transfer. Values of the physical 
properties of several propellants are given in Tables 8-5 and 7-1. 

2. When the wall temperature T„i exceeds the boiling point of the liquid by 
perhaps 10 to 50 K, small vapor bubbles form at the hot wall surface. 
These small, nuclei-like bubbles cause local turbulence, break away from 
the wall, and collapse in the cooler liquid. This phenomenon is known 
as nucleate boiling. The turbulence induced by the bubbles changes the 
character of the liquid film and, augmented by the vaporization of some 
of the propellant, the heat transfer rate is increased without a proportional 
increase in the temperature drop across the film, as can be seen by the steep 
slope B-C of the curve in Fig. 8-21. If the pressure of the fluid is raised, 
then the boiling point is also raised and the nucleate boiling region shifts to 
the right, to B'-C'. This boiling permits a substantial increase in the heat 
transfer beyond that predicted by Eq. 8-24. This phenomenon often occurs 
locally in the nozzle throat area, where the heat flux is high. The maximum 
feasible heat transfer rate (point C) is indicated as q^nx in Table 8-5 and 



*The grooves, tubes, or coolant passages in liquid propellant rocket chambers are often of complex 
cross section. The equivalent diameter, needed for fluid-film heat transfer calculations, is usually 
defined as four times the hydraulic radius of the coolant passage; the hydraulic radius is the cross- 
sectional flow area divided by the wetted perimeter. 



320 THRUST CHAMBERS 

appears to be a function of the cooling-fluid properties, the presence of 
dissolved gases, the pressure, and the flow velocity. In Table 8-5 it can 
be seen that hydrazine is a good heat absorber, but kerosene is poor (low 
^max and low critical pressure). 

3. As the heat transfer is increased further, the rate of bubble formation and 
the bubble size become so great that the bubbles are unable to escape from 
the wall rapidly enough. This reaction (shown as C-D in Fig. 8-21) is 
characterized by an unstable gas film and is difficult to obtain reproducibly 
in tests. When a film consisting largely or completely of gas forms along 
the hot wall surface, then this film acts as an insulation layer, causing a 
decrease in heat flux and, usually, a rapid increase in wall temperature, 
often resulting in a burnout or melting of the wall material. The cooling 
flow system has to be designed to avoid this unstable gas film regime 

4. As the temperature difference across the film is further increased, the wall 
temperatures reach values in which heat transfer by radiation becomes 
important. Region D-E is not of interest to cooling jacket designers. 

Cooling can also be accomplished by a fluid above its critical point with 
coolants such as hydrogen. In this case there is no nucleate boiling and the heat 
transfer increases with the temperature difference, as shown by the supercritical 
(dashed) line in Fig. 8-21. Liquid hydrogen is an excellent coolant, has a high 
specific heat, and leaves no residues. 

Chemical changes in the liquid can seriously influence the heat transfer from 
hot walls to liquids. Cracking of hydrocarbon fuel, with an attendant formation 
of insoluble gas, tends to reduce the maximum heat flux and thus promote failure 
more readily. Hydrocarbon fuel coolants (methane, jet fuel) can break down and 
form solid, sticky carbon deposits inside the cooling channel, impeding the heat 
transfer. Other factors influencing steady-state coolant heat transfer are gas radi- 
ation to the wall, bends in the coolant passage, improper welds or manufacture, 
and flow oscillations caused by turbulence or combustion unsteadiness. Some 
propellants, such as hydrazine, can decompose spontaneously and explode in the 
cooling passage if they become too hot. 

To achieve a good heat-absorbing capacity of the coolant, the pressure and 
the coolant flow velocity are selected so that boiling is permitted locally, but the 
bulk of the coolant does not reach this boiling condition. The total heat rejected 
by the hot gases to the surface of the hot walls, as given by Eq. 8-14 must be 
less than that permitted by the temperature rise in the coolant, namely 



qA = Q= mc(Ti - T 2 ) (8-25) 

where m is the coolant mass flow rate, c the average specific heat of the liquid, 
Ti the initial temperature of the coolant as it enters the cooling jacket, and T 2 its 
final temperature; Q is the rate of heat absorption per unit time; q is this same 
rate per unit heat transfer area A; T 2 should be below the boiling point prevailing 
at the cooling jacket pressure. 




CN CM ’-H (N ’-H 



0\ CM r^ 
^ 



cd 

u 

60 

^c 

'o 

P3 



3 

z 



° -a 

£ 



3 3 

[C Dh 









IT) IT) 

oooo-Hco-Hodo 

-H CM ^ CM CM 



o o o 

^ CM 




o^ 

00 

M3 



m 






r- 

o 

CM 



CM 

in 

in 



CM 


p 


CM 


CM 


O) 


CM 


i> 




CM 


»n 


I> 


l> 


00 


CM 


M3 O 


o 


CM 


o 


ON 


On 


00 


CM 


M3 O 


o 


m 




CM 


CM 


CM 


m 


<n cn 


cn 



Q- 

O 

u 

Oh 

'5 

cr 

J 



•c 



C3 

x: 

U 



I 

00 

U 

CQ 



c3 

Cl- 






00 ^ 
c ir 



U 



bJ) ^ 

.S 

^ Oh 
O C 

CQ g 



D 

V5 Cl 

V3 M 

a- 



o 

o 

U 



D 

cr 



i> p 

CM 



M3 

O 

M3 



CM 00 

in i> 

M3 M3 



^ CM 

m CM 

in 



r^inoooom-H-^^cM^ 

ooln'^ooo^Olnlno^'^o^ 

m-^inin'^'sO'O^cMcnm 



M3 o o^ 
cn o 00 
m ^ ^ 



-H o^ ^ On -h o^ ^ 

oooinONOooooooooomo 
-HNO^oo-HvomcO’-HNO^^ 



-H in 

o ^ 



o o cn NO o o 



o o ^ o -H cn 



T3 

X 






<u 



-a 








o 

t3 


c: 

p 


I 


o 


(U 




I I 

' ^ 


c 


c 


] X 


'n 


o 




! 




00 




I a 


"O 


o 


^ 




Vh 


3 ^ 


jz 


g 




I 





321 



322 THRUST CHAMBERS 



Radiation 

Radiation heat emission is the electromagnetic radiation emitted by a gas, liquid, 
or solid body by the virtue of its temperature and at the expense of its inter- 
nal energy. It covers the wavelength range from 10,000 to 0.0001 jam, which 
includes the visible range of 0.39 to 0.78 p.m. Radiation heat transfer occurs 
most efficiently in a vacuum because there is no absorption by the intervening 
fluids. 

The heat transmitted by the mechanism of radiation depends primarily on the 
temperature of the radiating body and its surface condition. The second law of 
thermodynamics can be used to prove that the radiant energy £ is a function of 
the fourth power of the absolute temperature T : 

E =fsaAT"^ (8-26) 

The energy E radiated by a body is defined as a function of the emissivity e, 
which is a dimensionless factor for surface condition and material properties, 
the Stefan -Boltzmann constant cr(5.67 x 10“^ W/m^-K"^), the surface area A, 
the absolute temperature T, and the geometric factor/, which depends on the 
arrangement of adjacent parts and the shape. At low wall temperatures (below 
800 K) radiation accounts for only a negligible portion of the total heat transfer 
in a rocket device and can usually be neglected. 

In rocket propulsion there are these radiation concerns: 

1 . Emission of hot gases to the internal walls of a combustion chamber, and its 
nozzle converging section, a solid propellant grain, or a hybrid propellant 
grain. 

2. Emission to the surroundings or to space from the external surfaces of 
hot hardware (radiation-cooled chambers, nozzles, or electrodes in electric 
propulsion). 

3. Radiation from the hot plume downstream of the nozzle exit. This is 
described in Chapter 20. 

In rocket combustion devices gas temperatures are between 1900 and 3900 K 
or about 3000 to 6600° F; their radiation contributes between 3 and 40% of the 
heat transfer to the chamber walls, depending on the reaction gas composition, 
chamber size, geometry, and temperature. It can be a significant portion of the 
total heat transfer. In solid propellant motors the radiation heating of the grain 
surfaces can be critical to the burning rate, as discussed in Chapter 12. The 
absorption of radiation on the wall follows essentially the same laws as those of 
emission. Metal surfaces and formed tubes reflect much of the radiant energy, 
whereas ablative materials and solid propellants seem to absorb most of the 
incident radiation. A highly reflective surface on the inside wall of a combustor 
tends to reduce absorption and to minimize the temperature increase of the walls. 

The hot reaction gases in rocket combustion chambers are potent radiation 
sources. Gases with symmetrical molecules, such as hydrogen, oxygen, and 



8.6. STARTING AND IGNITION 323 



nitrogen, have been found not to show many strong emission bands in those 
wavelength regions of importance in radiant heat transfer. Also, they do not 
really absorb much radiation and do not contribute considerable energy to the 
heat transfer. Heteropolar gases, such as water vapor, carbon monoxide, car- 
bon dioxide, hydrogen chloride, hydrocarbons, ammonia, oxides of nitrogen, and 
the alcohols, have strong emission bands of known wavelengths. The radiation of 
energy of these molecules is associated with the quantum changes in their energy 
levels of rotation and interatomic vibration. In general, the radiation intensity of 
all gases increases with their volume, partial pressure, and the fourth power of 
their absolute temperature. For small thrust chambers and low chamber pressures, 
radiation contributes only a small amount of energy to the overall heat transfer. 

If the hot reaction gases contain small solid particles or liquid droplets, then 
the radiation heat transfer can increase dramatically by a factor of 2 to 10. The 
particulates greatly increase the radiant energy as explained in Section 20.1. For 
example, the reaction gas from a gelled liquid propellants contains small solid 
particles and many solid propellants contain fine aluminum powder. When burned 
to form aluminum oxide, the heat of combustion and the combustion temperature 
are increased (raising heat transfer), and the specific impulse is raised somewhat 
(giving improved performance). The oxide can be in the form of liquid droplets 
(in the chamber) or solid particles (in the nozzle diverging section), depending on 
the local gas temperature. Furthermore, the impact of these particulates with the 
wall will cause an additional increase in heat transfer, particularly to the walls 
in the nozzle throat and immediately upstream of the nozzle throat region. The 
particles also cause erosion or abrasion of the walls. 



8.6. STARTING AND IGNITION 

The starting of a thrust chamber has to be controlled so that a timely and even 
ignition of propellants is achieved and the flow and thrust are built up smoothly 
and quickly to their rated value (see Refs. 6-1, 8-16, and 8-17). The initial 
propellant flow is always less than full flow , and the starting mixture ratio is 
usually different from the operating mixture ratio. A low initial flow prevents 
strong water hammer, gives a low initial heat release, and for nonhypergolic 
propellants it prevents an excessive accumulation of unignited liquid propellants 
in the chamber. 

The starting injection velocity is low, the initial vaporization, atomization, and 
mixing of propellants in a cold combustion chamber is often incomplete, and 
there are local regions of lean and rich mixtures. With cryogenic propellants the 
initial chamber temperature can be below ambient. The optimum starting mixture 
is therefore only an average of a range of mixture ratios, all of which should 
be readily ignited. Mixture ratios near the stoichiometric mixture ratio have a 
high heat release per unit of propellant mass and therefore permit bringing the 
chamber and the gases up to equilibrium faster than would be possible with other 
mixtures. The operating mixture ratio is usually fuel rich and is usually selected 



324 THRUST CHAMBERS 



for optimum specific impulse. One method of analytical modeling of the ignition 
of cryogenic propellants is given in Ref. 8-16. 

The time delay for starting a thmst chamber ideally consists of the following 
time periods: 

1. Time needed to fully open the propellant valves (typically 0.002 to more 
than 1.00 sec, depending on valve type and its size and upstream pressure). 

2. Time needed to fill the liquid passage volume between the valve seat and 
the injector face (piping, internal injector feed holes, and cavities). 

3. Time for forming discrete streams or sprays of liquid propellant (sometimes 
gaseous propellant, if cryogenic liquid is preheated by heat from cooling 
jacket) and for initial atomization into small droplets and for mixing these 
droplets. 

4. For today’s hypergolic propellant combinations at ambient initial tempera- 
tures, the combustion starts within a few milliseconds after a fuel droplet 
or fuel vapor comes into contact with an oxidizer droplet or oxidizer vapor. 
This delay can be longer at low ambient temperatures and at off-mixture 
ratios. 

5. For nonhypergolic propellant combinations an igniter system has to provide 
sufficient heat to bring a flow of mixed propellant to its ignition tempera- 
ture, before combustion can safely start. The igniter is usually started before 
any propellant is admitted to the chamber and usually before the propellant 
valves are opened. The time for the igniter to operate before any propellant 
comes in contact with the igniter flame can be one or more seconds in 
larger thrust chambers. If the igniter fails to operate and if this is sensed, 
then the engine controls can block the opening of the propellant valves. 

6. Time needed for droplets to vaporize and ignite (laboratory tests show this 
to be short, 0.02 to 0.05 sec, but this depends on the propellants and the 
available heat). 

7. Once ignition is achieved at a particular location in the chamber, it takes 
time to spread the flame or to heat all the mixed propellant that is entering 
into the chamber, vaporizing it, and raising it to ignition temperature. 

8. Time needed to raise the chamber pressure and temperature to the point 
where combustion will be self-sustaining. Then it is raised to its full pres- 
sure. 

There are overlaps in these delays and several of them can occur simultaneously. 
The delays are longer with large injectors or large-diameter chambers. Small 
thrusters can usually be started very quickly, in a few milliseconds, while larger 
units require 1 sec or more, sometimes as much as 5 sec. 

In starting a thmst chamber, one propellant always reaches the chamber a 
short time ahead of the other; it is almost impossible to synchronize exactly 
the fuel and oxidizer feed systems so that the propellants reach the chamber 
simultaneously at all injection holes or spray elements. Frequently, a more reliable 



8.6. STARTING AND IGNITION 325 



ignition is assured when one of the propellants is intentionally made to reach the 
chamber first. For example, for a fuel-rich starting mixture the fuel is admitted 
first. Reference 8-17 describes the control of the propellant lead. 

Other factors influencing the starting flows, the propellant lead or leg, and 
some of the delays mentioned above are the liquid pressures supplied to the 
injector (e.g., regulated pressure), the temperature of the propellant (some can be 
close to their vapor point), and the amount of insoluble gas (air bubbles) mixed 
with the initial quantity of propellants. 

The propellant valves (and the flow passages betwen the valves and the injec- 
tor face) are often so designed and controlled that they operate in a definite 
sequence, thereby assuring an intentional lead of one of the propellants and a 
controlled buildup of flow and mixture ratio. Often the valves are only partially 
opened because it is easier to ignite a small flow and it avoids an accumulation 
of hazardous unburned propellant mixture in the chamber. Once combustion is 
established, the valves are fully opened and full flow may reach the thrust cham- 
ber assembly. The initial reduced flow burning period is called the preliminary 
stage. Section 11.4 describes the starting controls. 

Full flow in the larger thrust chambers is not initiated with non- self-igniting 
propellants until the controller received a signal of successful ignition. The verifi- 
cation of ignition or initial burning is often built into engine controls using visual 
detection (photocell), heat detection (pyrometer), a fusible wire link, or sensing 
of a pressure rise. If the starting controls are not designed properly, unburnt 
propellant may accumulate in the chamber; upon ignition it may then explode, 
causing sometimes severe damage to the rocket engine. Starting controls and 
engine calibrations are discussed in Sections 11.4 and 11.5. 

Nonspontaneously ignitable propellants need to be activated by absorbing 
energy prior to combustion initiation. This energy is supplied by the ignition 
system. Once ignition has begun the flame is self-supporting. The igniter has to 
be located near the injector in such a manner that a satisfactory starting mixture 
at low initial flow is present at the time of igniter activation, yet it should not 
hinder or obstruct the steady-state combustion process. At least five different 
types of successful propellant ignition systems have been used. 

Spark plug ignition has been used successfully on liquid oxygen -gasoline 
and on oxygen -hydrogen thrust chambers, particularly for multiple starts during 
flight. The spark plug is often built into the injector, as shown in Fig. 9-6. 

Ignition by electrically heated wires has been accomplished but at times has 
proven to be less reliable than spark ignition for liquid propellants. 

Pyrotechnic ignition uses a solid propellant squib or grain of a few seconds’ 
burning duration. The solid propellant charge is electrically ignited and bums 
with a hot flame within the combustion chamber. Almost all solid propellant 
rockets and many liquid rocket chambers are ignited in this fashion. The igniter 
container may be designed to fit directly onto the injector or the chamber (see 
Fig. 8-1), or may be held in the chamber from outside through the nozzle. This 
ignition method can only be used once; thereafter the charge has to be replaced. 



326 THRUST CHAMBERS 



Ignition has been achieved in a precombustion chamber also called premix 
chamber; see Fig. 8-3; it is a small chamber built next to the main combustion 
chamber and connected through an orifice; this is similar to the precombustion 
chamber used in some internal combustion engines. A small amount of fuel and 
oxidizer is injected into the precombustion chamber and ignited. The burning 
mixture enters the main combustion chamber in a torchlike fashion and ignites 
the larger main propellant flow which is injected directly into the main chamber. 
This ignition procedure permits repeated starting of thrust chambers and has 
proved successful with the liquid oxygen -gasoline and oxygen -hydrogen thrust 
chambers. 

Auxiliary fluid ignition is a method whereby some hypergolic liquid or gas, in 
addition to the regular fuel and oxidizer, is injected into the combustion chamber 
for a very short period during the starting operation. This fluid is hypergolic, 
which means it produces spontaneous combustion with either the fuel or the oxi- 
dizer. The combustion of nitric acid and some organic fuels can, for instance, 
be initiated by the introduction of a small quantity of hydrazine or aniline at the 
beginning of the rocket operation. Liquids that ignite with air (zinc diethyl or alu- 
minum triethyl), when preloaded in the fuel piping, can accomplish a hypergolic 
ignition. The flow diagram of the RD 170 Russian rocket engine in Fig. 11-2 
shows several cylindrical containers prefilled with a hypergolic liquid, one for 
each of the high-pressure fuel supply lines; this hypergolic liquid is pushed out 
(by the initial fuel) into the thrust chambers and into the prebumers to start their 
ignitions. 

In vehicles with multiple engines or thrust chambers it is required to start two 
or more together. It is often difficult to get exactly simultaneous starts. Usually 
the passage or manifold volumes of each thrust chamber and their respective 
values are designed to be the same. The temperature of the initial propellant 
fed to each thrust chamber and the lead time of the first quantity of propellant 
entering into the chambers have to be controlled. This is needed, for example, 
in two small thrusters when used to apply roll torques to a vehicle. It is also 
one of the reasons why large space launch vehicles are not released from their 
launch facility until there is assurance that all the thrust chambers are started and 
operating. 



8.7. RANDOM VARIABLE THRUST 

Only some applications require a randomly variable thrust from an engine. 
Examples are descend on a planet or the moon, combat aircraft or upper stage 
of an anti-ballistic missile. One of the advantages of liquid propellant rocket 
engines is that they can be designed to throttle or randomly vary its thrust over 
a wide range during flight. There have been several approaches to achieve vari- 
able or randomly reduced thrust. Essentially all the approaches to vary the thrust 
involve the thrust chamber and therefore this section has been placed into the 
thrust chamber chapter. One of the early schemes was to use multiple engines or 



8.7. RANDOM VARIABLE THRUST 327 



multiple thrust chambers on the same engine and then stop the operation of one 
or more of them. The Reaction Motors 6,000C-4 engine had four thrust chambers 
and they could be turned on or off individually; this gave a step change in thrust. 

Today one can distinguish between two modes of operation and both depend 
on reducing the propellant flow. The first is moderate throttling, typically over 
a thrust range of two or three, and this can usually be accomplished without 
signihcant design changes to the engine. In one application the thrust is throttled 
during the ascent of a booster vehicle in order to prevent excessive aerodynamic 
heating. The other mode, often called deep throttling, varies the thrust by a factor 
of between 6 to 30 for specific engines. It applies, for example, to a planetary 
landing rocket engine with controlled deceleration. To achieve this deep throttling, 
the engine requires some special features and some of these are mentioned below. 

The thrust is almost proportional to the propellant mass flow (Eq. 2-14) and 
therefore reducing that flow will reduce the thrust. With moderate throttling it 
can be achieved by simultaneously closing the main fuel valve and the main 
oxidizer valve (with matching flow characteristics) or alternatively by slowing 
down the rotary speed of the turbopumps (by reducing the gas flow to the turbine) 
with a hydraulically matched fuel pump and oxidizer pump. At lower flow the 
pressure drop across the injector and the injection velocity will be diminished, 
the atomization and mixing of the propellants will be somewhat less efficient and 
the combustion will be less complete, and the chamber pressure will be reduced. 
This usually causes a decrease in the specific impulse at the lower thrust levels 
of perhaps 1.5 to 9%, depending on the specific engine design. This method 
of varying the thrust is used in a number of booster engines for space launch 
vehicles. 

With deeper throttling the oxidizer and fuel flows tend to oscillate and will 
usually no longer be at the original mixture ratio, the propellant injection streams 
or sprays and their impingement locations will wander, and the flow is likely to 
become erratic. To prevent these phenomena the engines have some special fea- 
tures as illustrated by the lunar landing rocket engine; this engine was throttled 
by a factor of up to 10:1, was developed by the predecessor of Northrop Grum- 
man, and flew in the late 1960s and 1970s. Its propellant flow control valves 
each included a cavitating venturi with a movable tapered pintle that allowed to 
vary the flow area of the venturi throat of the thrust chamber valve. This assured 
a predetermined steady reduced flow of propellants at all thrust levels, and main- 
tained the mixture ratio to be the same as the full thrust value. Furthermore the 
injection of the propellants was through a pintle injector with two annular slots 
of a moving sleeve and the width of these slots could be reduced by an actuator 
built into the injector. The lower right sketch of Fig. 8-3 shows this feature. It 
allows high injection velocities of the liquid propellants, giving good atomization 
and pretty good combustion and a relatively small loss in performance at low 
thrust. The highest thrust variation of 357 to 1.0 was with the sustainer engines of 
the Fance surface-to-surface missile. Its engine, developed by Rocketdyne (today 
Pratt & Whitney Rocketdyne), hrst flew in the late 1960s. The specific impulse 
at low thrust was poor (more than 15% loss). 



328 THRUST CHAMBERS 



Another approach to achieve variable thrust is to also vary the nozzle throat 
area. See Eq. 3-31. This scheme requires a moveahle tapered pintle in the main 
nozzle throat area. The pintle has to he made of heat-tolerant material or has to he 
regeneratively cooled and the pintle position is usually hydraulically controlled. 
This will allow to maintain essentially constant chamber pressure at all thrust 
levels. Several experimental liquid propellant engines and also solid propellant 
motors with a random variable nozzle throat area have been built and tested. To 
the best of the author’s knowledge, none have flown. 

For small reaction control thrusters the average thrust is usually reduced 
by pulsing. It is accomplished by controlling the number of cycles or pulses 
(each has one short fixed-duration thrust pulse plus a short fixed-duration zero- 
thrust pause), by modulating the duration of individual pulses (with short pauses 
between pulses), or alternatively by lengthening the pause between pulses. 



8.8. SAMPLE THRUST CHAMBER DESIGN ANALYSIS 

This example shows how a thrust chamber is strongly influenced by the overall 
vehicle system requirements or the mission parameters and the vehicle design. 
As outlined in the design section of Chapter If and in the discussion of the 
selection of propulsion systems in Chapter 19, each engine goes through a series 
of rationalizations and requirements that define its key parameters and its design. 
In this example we describe one way of how the thrust chamber parameters are 
derived from the vehicle and engine requirements. The overall system require- 
ments relate to the mission, its purpose, environment, trajectories, reusability, 
reliability, and to restraints such as allowable engine mass, or maximum dimen- 
sional envelope. We are listing some, but not all of the requirements. It shows 
how theory is blended with experience to arrive at the initial choices of the design 
parameters. This example could be different, if it would be done by a different 
design team. 

Example 8-2. Here we define the application as a new upper stage of an existing 
multistage space launch vehicle that will propel a payload into deep space. This means 
continuous firing (no restart or reuse), operating in the vacuum of space (high nozzle area 
ratio), modest acceleration (not to exceed 5 go), reasonably low cost, moderately high 
performance (specific impulse), and a thrust whose magnitude depends on the payloads, 
the flight path, and acceleration limits. The desired mission velocity increase of the stage 
is 3400 m/sec. The engine is attached to its own stage, which is subsequently disconnected 
and dropped from the payload stage. The payload stage (3500 kg) consists of a payload of 
1500 kg (for scientific instruments, power supply, or communications and flight control 
equipment) and its own propulsion systems (including propellant) of 2000 kg (for trajec- 
tory changes, station keeping, attitude control, or emergency maneuvers). There are two 
geometric restraints: The vehicle has an outside diameter of 2.0 m, but when the structure, 
conduits, certain equipment, thermal insulation, fittings, and assembly are considered, it 



8.8. SAMPLE THRUST CHAMBER DESIGN ANALYSIS 329 



really is only about 1.90 m. The restraint on the stage length of 4.50 m maximum will 
affect the length of the thrust chamber. We can summarize the key requirements: 



Application 

Payload 

Desired velocity increase Au 
Maximum stage diameter 
Maximum stage length 
Maximum acceleration 



Uppermost stage to an existing 
multistage launch vehicle 
3500 kg 

3400 m/sec in gravity-free vacuum 
1.90 m 
4.50 m 
5 go 



Decisions on Basic Parameters. The following engine design decisions or 
parameter selection should he made early in the design process: 

Propellant combination 
Chamber pressure 
Nozzle area ratio 

Feed system, using pumps or pressurized tanks 
Thrust level 

From a performance point of view, the best propellant combination would 
be liquid oxygen with liquid hydrogen. Flowever, this bipropellant would have 
a low average specific gravity (0.36). There is not enough volume in this upper 
stage to allow sufficient propellant mass for attaining the desired vehicle velocity 
increment Am. The lower stages of the existing launch vehicle use liquid oxygen 
with RP-1 fuel with an average specific gravity of about 1.014, and the launch 
pad is already equipped for supplying these. The new stage is limited in volume 
and cross section. Because of these factors the propellant combination of liquid 
oxygen and RP-1 (a type of kerosene) is selected. From Fig. 5-1 we see that 
the theoretical specific impulse is between 280 and 300 sec, depending on the 
mixture ratio and whether we use frozen or shifting chemical equilibrium in 
the nozzle flow expansion. This figure also shows that the maximum value of 
the characteristic velocity c* is reached at a mixture ratio of about 2.30, which 
is a fuel-rich mixture. We select this mixture ratio. Its combustion temperature 
is lower than the mixture ratios with higher values, and this should make the 
cooling of the thrust chamber easier. We will see later that cooling may present 
some problems. Based on universal experience, we select a value of 4 part way 
(about 40%) between the values for frozen and shifting equilibrium, namely 292 
sec at the standard chamber pressure of 1000 psi or 6.895 MPa, and a nozzle big 
enough for expansion to sea level. From Fig. 5-1 and Table 5-5 we find the 
molecular mass to be 23 kg/kg-mol and the specific heat ratio k to be about 1 .24. 
Later we will correct this value of 4 from this standard reference condition to 
the actual vacuum specific impulse of the thrust chamber with its actual nozzle 
exit area ratio. 



330 THRUST CHAMBERS 



Next we will select a chamber pressure, a nozzle area ratio, and a feed system 
concept. Historically there has been favorable experience with this propellant 
combination at chamber pressures between 100 and 3400 psia with nozzle area 
ratios up to about 40 with both gas generator cycles and staged combustion 
cycles, giving proof that this is feasible. The following considerations enter into 
this selection: 

1. Higher chamber pressures allow a smaller thrust chamber and (for the 
same nozzle exit pressure) a shorter nozzle cone with a smaller nozzle exit 
diameter. The thrust chamber is small enough for a toroidal tank to be built 
around it, and this conserves stage length. This not only saves vehicle space 
but usually also some inert mass in the vehicle and the engine. Figure 8-22 
shows the relative sizes of thrust chambers for three chamber pressures 
and two nozzle area ratios (e of 100 and 300). The nozzle length and exit 
diameter cannot exceed the values given in the requirements, which, as can 
be seen, rules out low chamber pressure or a large nozzle. The dimensions 
shown are calculated later in this analysis. 

2. The heat transfer rate, which varies with the gas-film coefficient, is almost 
proportional to the gas density, which is nearly proportional to the chamber 
pressure, as shown by Eq. 8-20 or 8-22. On some prior thrust cham- 
bers there have been problems with the formation of solid carbon layer or 
deposits either inside the cooling jacket (increasing wall temperatures) or 
on the inner walls of the combustion chamber (the solid can flake off and 
cause burnout). This favors a lower chamber pressure. 



Pi = 8.962 MPa p, = 4.826 MPa p^ = 0.689 MPa 

(1300 psia) (700 psia) (100 psia) 




FIGURE 8-22. Comparison of thrust chamber sizes for three chamber pressures and 
two nozzle area ratios (100 and 300). 



8.8. SAMPLE THRUST CHAMBER DESIGN ANALYSIS 331 

3. Concern over leak-free seals for both static and dynamic seals increases 
with chamber pressure, which in turn causes all feed pressures also to 
increase. 

4. A feed system using pressurized gas is feasible, but its inert masses of tanks 
and engine are favorable only if the chamber pressure is very low, perhaps 
around 100 psia or less. The tanks for propellants and pressurizing gas 
become very heavy and the thrust chamber will be very large and exceed 
the dimensional restraints mentioned above. We therefore cannot use this 
feed system or very low chamber pressures. 

5. If we use a pump feed system, the power needed to drive the pumps 
increases directly with chamber pressure p\. In a gas generator engine 
cycle this means a slightly reduced performance as the value of p\ goes 
up. For a staged combustion cycle it means high pressures, particularly 
high-pressure hot gas flexible piping, and a more complex, heavier, and 
expensive engine. We therefore select a gas generator cycle (see Fig. 1-4) 
at a low enough chamber pressure so that the thrust chamber (and the other 
inert hardware) will just fit the geometrical constraints, and the engine inert 
mass and the heat transfer will be reasonable. 

For these reasons we pick a chamber pressure of 700 psia or 4.825 MPa and an 
area ratio of 100. With further analysis we could have picked p\ more precisely; 
it could be slightly lower. Next we correct the specific impulse to the operating 
conditions using a ratio of thrust coefficients. We can use Eq. 3-30 or interpolate 
between Figs. 3-7 and 3-8 for a value of k = 1.24. The reference or standard 
condition (see Fig. 3-6) is for a pressure ratio pi/p$ of 1000/14.7 = 68, which 
corresponds to an area ratio of about 8. Then (Cf)standard = 1-58. For the actual 
high-altitude operation the pressure ratio is close to inhnity. The nozzle has an 
area ratio of 100; we can determine the thrust coefficient by interpolating with 
pressure at k = 1.24. The result is (Cf)vacuum = 1-90. The new ideal specific 
impulse value for a chamber threshold of 700 psia and a nozzle area ratio of 
100 is therefore 292 x (1.90/1.58) = 351.1 sec. In order to correct for losses 
(divergence, boundary layer, incomplete combustion, some film cooling, etc.) 
we use a correction factor of 0.96 giving a thrust chamber specific impulse of 
337.1 sec. The engine uses a gas generator and this will reduce the engine specific 
impulse further by a factor of approximately 0.98 or (/i)engine = 330.3 sec or an 
effective exhaust velocity of 3237 m/sec. 

Stage Masses and Thrust Level. An estimate of the stage masses will next 
be made. We assume that the inert hardware (tanks, gas, generator, turbopumps, 
etc.) is about 7% of the propellant mass, which is conservative when compared 
to existing engines. In a full-fledged engine design this number would be verified 
or corrected once an estimated but more detailed mass budget becomes available. 
From Eq. 4-7 

Au/v _ '^p + 0-07OTp + 3500 

^ ~~mf~ 007^7+3500 



3400/3237 



332 THRUST CHAMBERS 



Solve for nip — 7483 kg. The final and initial masses ntf and mo of the stage are 
then 4023 kg and 11,510 kg, respectively. 

The maximum thrust is limited by the maximum allowed acceleration of 
5go- It is Tmax = tUQU = 11,510 X 5 X 9.8 = 564,400 N. This would become 
a relatively large and heavy thrust chamber. Considerable saving in inert mass 
can be obtained if a smaller thrust size (but longer firing duration) is chosen. 
Since this same thrust chamber is going to be used for another mission where an 
acceleration of somewhat less than 1.0 go is wanted, a thrust level of 50,000 N or 
11,240 Ibf is chosen. The maximum acceleration of the stage occurs just before 
cutoff; it is a = F/mf = 50,000/4023 = 12.4 m/sec^ or about 1.26 times the 
acceleration of gravity. This fits the thrust requirements. 

The following have now been determined: 



Propellant 

Mixture ratio (0/F) 

Thrust 

Chamber pressure 
Nozzle area ratio 
Specific impulse (engine) 

Specific impulse (thrust chamber) 

Engine cycle 

Usable propellant mass 

Estimated nozzle exit exhaust velocity 



Liquid oxygen and liquid kerosene 
(RP-1) 

2.30 (engine) 

50,000 N or 11,240 Ibf 
700 psia or 4.826 MPa 
100 

330.3 sec 
337.1 sec 
Gas generator 
7478 kg 

3237 m/sec or 10,613 ft/sec 



Propellant Flows and Dimensions of Thrust Chamber. From Eq. 2-6 we 

obtain the propellant mass flow: 



m=F/c = 50,000/3237 = 15.446 kg/sec 

When this total flow and the overall mixture ratio are known, then the fuel flow 
rhf and oxidizer flow tko for the engine, its gas generator, and its thrust chamber 
can be determined from Eqs. 6-3 and 6-4 as shown below: 



riif = OT/(r + 1) = 15.446/(2.3 + 1) = 4.680 kg/sec 

tho = mr/{r + 1) = (15.446 x 2.30)/3.30 = 10.765 kg/sec 

The gas generator flow rhgg consumes about 2.0% of the total flow and operates 
at a fuel-rich mixture ratio of 0.055; this results in a gas temperature of about 
890 K: 



(^/)gg = 0.2928 kg/sec (^o)gg = 0.0161 kg/sec 

The flows through the thrust chamber are equal to the total flow diminished by 
the gas generator flow, which is roughly 98.0% of the total flow or 15.137 kg/sec: 



(mf )tc = 4.387 kg/sec 



(Wo)tc = 10.749 kg/sec 



8.8. SAMPLE THRUST CHAMBER DESIGN ANALYSIS 333 



The duration is the total effective propellant mass divided by the mass flow 
rate: 

tj, = nip/thp = 7478/15.446 = 484.1 sec or a little longer than 8 min 

The nozzle throat area is determined from Eq. 3-31. This neglects start and stop 
transients: 

At = F/{PiCf) = 50,000/(4.826 x 10® x 1.90) = 0.005453 m^ or 54.53 cm^ 

The nozzle throat diameter is Z), = 8.326 cm. The internal diameter of the 
nozzle at exit A 2 is determined from the area ratio of 100 to be D 2 = x/TOO x Dj 
or 83.26 cm. A shortened or truncated bell nozzle (as discussed in Section 3.4) 
will be used with 80% of the length of a 15° conical nozzle, but with the same 
performance as a 15° cone. The nozzle length (from the throat to the exit) can 
be determined by an accurate layout or by an equivalent 15° conical nozzle exit 
L = (D 2 — Dt)/(2 tan 15) as 139.8 cm. For an 80% shortened bell nozzle this 
length would be about 111.8 cm. The contour or shape of a shortened bell nozzle 
can be approximated by a parabola (parabola equation is = 2px). Using an 
analysis (similar to the analysis that resulted in Fig. 3-14), the maximum angle 
of the diverging section at the inflection point would be about 6>, = 34° and the 
nozzle exit angle 9e = 7°. The approximate contour consists of a short segment 
of radius 0.4r( of a 34° included angle (between points T and I in Fig. 3-14) and 
a parabola with two known points at / and E. Knowing the tangent angles (34 
and 7°) and the y coordinates [y^ = r 2 and y, = r, + 0.382 r,(l — coi0,)] allows 
the determination of the parabola by geometric analysis. Before detail design 
is undertaken, a more accurate contour, using the method of characteristics, is 
suggested. 

The chamber diameter should be about twice the nozzle throat diameter to 
avoid pressure losses in the combustion chamber {E>c = 16.64 cm). Using the 
approximate length of prior successful smaller chambers and a characteristic 
length L* of about 1.1m, the chamber length (together with the converging nozzle 
section) is about 11.8 in. or 29.9 cm. The overall length of the thrust chamber 
(169 cm) is the sum of the nozzle length (1 1 1.8 cm), chamber (29.9 cm), injector 
thickness (estimated at 8 cm), mounted valves (estimated at 10 cm), a support 
structure, and possibly also a gimbal joint. The middle sketch of the three thrust 
chambers in Fig. 8-22 corresponds roughly to these numbers. 

We have now the stage masses, propellant flows, nozzle, and chamber config- 
uration. Since this example is aimed at a thrust chamber, data on other engine 
components or parameters are given only if they relate directly to the thrust 
chamber or its parameters. 

Next we check if there is enough available vehicle volume (1.90 m diameter 
and 4.50 m long) to allow making a larger nozzle area ratio and thus gain a little 
more performance. First we determine how much of this volume is occupied 
by propellant tanks and how much might be left over or be available for the 



334 THRUST CHAMBERS 



thrust chamber. This analysis would normally be done by tank design specialists. 
The average density of the propellant mixture can be determined from Eq. 7-1 
to be 1014 kg/m^ and the total usable propellant of 7478 kg. Using densities 
from Table 7-1 the fuel volume and the oxidizer volume can be calculated to be 
2.797 and 4.571 m^, respectively. For a diameter of 1.90 m, a nearly spherical 
fuel tank, a separate oxidizer cylindrical tank with elliptical ends, 6% ullage, and 
2% residual propellant, a layout would show an overall tank length of about 3.6 
m in a space that is limited to 4.50 m. This would leave only 0.9 m for the length 
of the thrust chamber, and this is not long enough. Therefore we would need to 
resort to a more compact tank arrangement, such as using a common bulkhead 
between the two tanks or building a toroidal tank around the engine. It is not 
the aim to design the tanks in this example, but the conclusion affects the thrust 
chamber. Since the available volume of the vehicle is limited, it is not a good 
idea to try to make the thrust chamber bigger. 

This diversion into the tank design shows how a vehicle parameter affects 
the thrust chamber design. For example, if the tank design would turn out to be 
difficult or the tanks would become too heavy, then one of these thrust chamber 
options can be considered: (1) go to a higher chamber pressure (makes the thrust 
chamber and nozzle smaller, but heavier), (2) go to a lower thrust engine (will 
be smaller and lighter), (3) store the nozzle of the upper-stage thrust chamber in 
two pieces and assemble them during the flight once the lower stages have been 
used and discarded (see extendible nozzle in Fig. 8-17; it is more complex and 
somewhat heavier), or (4) use more than one thrust chamber in the engine (will 
be heavier, but shorter and can provide roll control). We will not pursue these or 
other options here. 

Heat Transfer. The particular computer program for estimating heat transfer 
and cooling parameters of thrust chambers will depend on the background and 
experience of specific engineers and rocket organizations. Typical computer pro- 
grams divide the internal wall surface of the chamber and nozzle into incremental 
axial steps. Usually in a preliminary analysis the heat transfer is estimated only 
for critical locations such as for the throat and perhaps the chamber. 

From Fig. 5-1 and Eq. 3-12 or 3-22 we determine the following gas temper- 
atures for the chamber, nozzle throat region, and a location in the diverging exit 
section. They are: T\ — 3600 K, T, = 3214 K, and = 1430 K at an area ratio 
of 6.0 in the diverging nozzle section. The chamber and nozzle down to an exit 
area ratio of 6 will have to be cooled by fuel. For this propellant combination and 
for the elevated wall temperatures a stainless steel has been successfully used for 
the inner wall material. 

Notice that beyond this area ratio of about 6, the nozzle free-stream gas tem- 
peratures are relatively low. Uncooled high-temperature metals can be used here 
in this outer nozzle region. Radiation cooling, using a material such as niobium 
(coated to prevent excessive oxidation) or carbon fibers in a nonporous carbon 
matrix, is suitable between an area ratio of 6 and about 25. For the final large 
nozzle exit section, where the temperatures are even lower, a lower cost material 



8.8. SAMPLE THRUST CHAMBER DESIGN ANALYSIS 335 



such as stainless steel or titanium is suggested. Ablative materials have been ruled 
out because of the long duration and the aggressive ingredients in the exhaust 
gas. The gas compositions of Figs. 5-2 and 5-3 indicate that some free oxygen 
and hydroxyl are present. 

We now have identified the likely materials for key chamber components. The 
best way to cool the radiation-cooled exit segment of the nozzle (beyond area 
ratio of 6) is to let it stick out of the vehicle structure; the heat can then be freely 
radiated to space. One way to accomplish this is to discard the vehicle structure 
around the nozzle end after stage separation and before second-stage start. 

As in Fig. 8-8, the maximum heat transfer rate will be at the nozzle throat 
region. A variety of heat transfer analysis programs are available for estimat- 
ing this heat transfer. If a suitable computer program is not available, then an 
approximate steady-state heat transfer analysis can be made using Eqs. 8-14 to 
8-18 and the physical properties (specific heat, thermal conductivity, and den- 
sity) of RP-1 at elevated temperatures. The film coefficients of Eqs. 8-22 and 
8-24 are also needed. This is not done in this example, in part because data 
tables for the physical properties would take up a lot of space and results are 
not always reliable. Data from prior thrust chambers with the same propellants 
indicate a heat transfer rate at the nozzle throat region exceeding 10 Btu/in.^-sec 
or 1.63 X 10^ W/m^. 

The RP-1 fuel is an unusual coolant since it does not have a distinct boiling 
point. Its composition is not consistent and depends on the oil stock from which 
it was refined and the refining process. It is distilled or evaporated gradually 
over a range of temperatures. The very hot wall can cause the RP-1 to locally 
break down into carbon-rich material and to partially evaporate or gasify. As 
long as the small vapor bubbles are recondensed when they are mixed with the 
cooler portions of the coolant flow, a steady heat transfer process will occur. If 
the heat transfer is high enough, then these bubbles will not be condensed, may 
contain noncondensable gases, and the flow will contain substantial gas bubbles 
and become unsteady, causing local overheating. The recondensing is aided by 
high cooling passage velocities (more than 10 m/sec at the throat region) and by 
turbulence in these passages. A coolant flow velocity of 15 m/sec is selected for 
the nozzle throat region (adequate to prevent big gas bubbles), 7 m/sec for the 
chamber region and somewhat less, 3 m/sec for the cooled nozzle exit segment. 

The material for the cooling jacket will be stainless steel to resist the oxidation 
and erosion of the fast moving, aggressive hot gas, which contains a small amount 
of free oxygen and hydroxyl species. The forced cooling by fuel will assure that 
the temperatures of this stainless steel are well below its softening temperature 
of about 1050 K. 

The construction of the cooling jacket can be tubular, as shown in Figs. 8-1 
and 8-9, or it can consist of milled channels as shown in Figs. 8-2 and 8-15. 
The cross section of each tube or cooling channel will be a minimum at the throat 
region, gradually become larger, and be about two or more times as large at the 
chamber and diverging nozzle regions. The wall thickness (on the hot gas side) 
should be as small as possible to reduce the temperature drop across the wall 



336 THRUST CHAMBERS 



(which reduces the thermal stresses and allows a lower wall temperature) and to 
minimize the yielding of the material that occurs due to thermal deformation and 
pressure loads. Figure 8-11 shows this behavior, but for a thick wall. Practical 
considerations such as manufacturability, the number of test firings before flight, 
the deformation under pressure loads, the temperature gradient and dimensional 
tolerances also enter into the selection of the wall thickness. A inner wall thick- 
ness of 0.5 mm and a cooling velocity of 15 m/sec have been selected for the 
throat region of the cooling jacket. Milled slots (rather than tubes) have been 
selected for this thrust chamber. 

The selection of the number of milled slots, their cross sections, and the wall 
thickness is a function of the coolant mass flow, its pressure, wall stresses, wall 
material, and the shape of the channel. Figure 8-23 and Table 8-6 describe 
the channel width and height for different numbers of channels and different 
locations. The fuel coolant flow is diminished by the gas generator fuel flow 
(0.293 kg/sec) and is about 4.387 kg/sec. For this flow and a cooling velocity 
of 15 m/sec in the throat region the cumulative cross-sectional area of all the 
channels is only about 3.62 cm^. The cooling velocity is lower in the chamber 
and nozzle regions and the cumulative channel flow area will be larger there. 
The variables are the number of channels, the thickness of the hot wall, the rib 
thickness between channels, the cooling velocity, the gas temperature, and the 
location along the thrust chamber profile. The number of channels or tubes will 
determine the shape of the cross section, ranging from deep and thin to almost 
square. The effect of varying the number of channels or channel dimensions and 
shape is shown in Table 8-6. The minimum inert mass of the cooling jacket and 
a low friction loss occur, when the shape (which varies axially throughout the 
jacket) is on the average close to a square. On the basis of analyses, as shown 
in the table, a 150-channel design has been selected for giving favorable cross 
section, reasonable dimensions for ease of fabrication, good cooling, and often 
low thermal wall stresses. 

Reinforcing bands have to be put on the outside of the tubes or channels to 
hold the internal gas pressure during operation, to contain the coolant pressures, 
which cause heated walls wanting to become round, and any surge pressures 
during the start transient or arising from water hammer in the lines. We assume a 
surge pressure of 50% above chamber pressure and a steel strength of 120,000 psi. 
In the chamber the inside diameter is 16.7 cm (6.57 in.), the walls and channels 




Hot gas side 
of inner wall 



Wall 

thickness 



FIGURE 8-23. Segment of cooling jacket with milled channels and an electroformed 
outer wall. 



8.8. SAMPLE THRUST CHAMBER DESIGN ANALYSIS 337 



TABLE 8-6. Alternative Milled Channel Configurations for Fuel (Cooling) Flow of 
4.387 kg/sec 



Throat Section 






Chamber Section 






Wall thickness 


0.05 cm 


Wall thickness 


0.06 cm 


Rib thickness 


0.08 cm 


Rib thickness 


0.08 




Total flow area 


3.653 cm^ 


Total flow area 


7.827 cm2 


Flow velocity 


15 m/sec 


Flow velocity 


7.0 m/sec 


Number of 


Channel 


Channel 


Number of 


Channels 


Channel 


Channels 


Width, cm 


Depth, cm 


Channels 


Width, cm 


Depth, cm 


80 


0.257 


0.177 








100 


0.193 


0.189 


100 


0.456 


0.171 


120 


0.145 


0.210 


120 


0.367 


0.179 


140 


0.113 


0.231 


140 


0.303 


0.184 


150 


0.100 


0.243 


150 


0.277 


0.188 


160 


0.092 


0.247 


160 


0.255 


0.192 


180 


0.070 


0.289 


180 


0.218 


0.196 



each are 0.3 cm deep, and the pressure is 700 psia or 4.826 MPa. If one band 
allows the reinforcing of a length of chamber of 3.0 in., the cross-sectional area 
of that reinforcing band will be 

A = pDLjila) = [700 x 1.5 x (6.57 -F 0.3) x 3]/(2 x 120,000) = 0.0902 in.^ 



If the bands were 1.0 in. wide, its thickness would be 0.09 in.; and if it were 3 
in. wide, it would be 0.03 in. thick. Large nozzle exit sections have been observed 
to experience flutter or cyclic deformation, and therefore some stiffening rings 
may be needed near the exit. 

The capacity of the fuel to absorb heat (Eq. 8-25) is approximately 
CpWi/ AT = 0.5 X 4.81 x 200 = 278,000 J/sec. The maximum AT is estab- 
lished by keeping the fuel well below its chemical decomposition point. This 
calculated heat absorption is less than the heat transfer from the hot gases. 
It is therefore necessary to reduce the gas temperature near the chamber and 
nozzle walls or to increase the heat absorption. This can be accomplished by 

(1) introducing film cooling by injection into the chamber just ahead of the 
nozzle, by (2) modifying the injection patterns, so that a cooler, fuel-rich thick 
internal boundary layer is formed, or (3) by allowing some nucleate boiling in 
the throat region. The analysis of these three methods is not given here. Item 

(2) , supplementary cooling, is selected because it is easy to design and build 
and can be based on extensive data of prior favorable experience. However, it 
causes a small loss of performance (up to about 1% in specific impulse). The 
specific impulse of the engine is then 327 sec. 



338 THRUST CHAMBERS 



Injector Design. The injector pattern can be any one of the several types shown 
in Figs. 8-3 and 8-4. For this propellant combination we have used both doublets 
(like and unlike), and triplets in the United States, and the Russians have used 
multiple hollow double posts with swirling or rotation of the flow in the outer 
annulus. Based on good experience and demonstrated combustion stability with 
similar designs, we select a doublet self-impinging-type stream pattern (Fig. 8-3) 
and an injector structure design similar to Fig 8-5. The impinging streams form 
fans of liquid propellant, which break up into droplets. Oxidizer and fuel fans 
alternate radially. We could also use a platelet design, like Fig. 8-6. 

The pressure drop across the injector is usually set at values between 15 and 
25% of the chamber pressure, in part to obtain high injection velocities, which 
aid in atomization and droplet breakup. In turn this leads to more complete 
combustion (and thus better performance) and to stable combustion. We will 
use 20% or 140 psi or 0.965 MPa for the injector pressure drop. There is a 
small pressure loss in the injector passages. The injection velocities are found 
from Eqs. 8-1 and 8-5. The equation is solved for the area A, which is the 
cumulative cross-section area of all the injection holes of one of the propellants 
in the injector face. 

With rounded and clean injection hole entrances the discharge coefficient will 
be about 0.80 as shown in Table 8-2. Solving for the cumulative injection hole 
area for the fuel and the oxidizer flows gives 1.98 cm^ for the fuel and 4.098 cm^ 
for the oxidizer. A typical hole diameter in this size of injector would be about 
0.5 to 2.5 mm. We will use a hole size of 1.5 mm for the fuel holes (with 
90% of the fuel flow) and 2.00 mm for the oxidizer hole size, resulting in 65 
doublets of oxidizer holes and 50 doublets of fuel. By using a slightly smaller fuel 
injection hole diameter, we can match the number of 65 doublets as used with 
the oxidizer holes. These injection doublets will be arranged on the injector face 
in concentric patterns similar to Fig. 8-5. We may be able to obtain a slightly 
higher performance by going to smaller hole sizes and a large number of fuel 
and oxidizer holes. In addition there will be extra fuel holes on the periphery 
of the injector face to help in providing the cooler boundary layer, which is 
needed to reduce heat transfer. They will use 10% of the fuel flow and, for a 
0.5-mm hole diameter, the number of holes will be about 100. To make a good 
set of liquid fans, equal inclination angles of about 25° are used with the doublet 
impingements. See Fig. 8-7. 



Igniter Dimensions. A pyrotechnic (solid propellant) igniter will be used. It 
has to have enough energy and run long enough to provide the pressure and 
temperature in the thrust chamber for good ignition. Its largest diameter has to 
be small enough to be inserted through the throat, namely 8.0 cm maximum 
diameter and it can be 10 to 15 cm long. The actual igniter will, most likely, be 
smaller than this. 



PROBLEMS 339 



Layout Drawings, Masses, Flows, and Pressure Drops. We now have 
enough of the key design parameters of the selected thrust chamber, so a pre- 
liminary layout drawing can be made. Before this can be done well, we will 
need some analysis or estimates on the thrust chamber manifolds for fuel and 
oxidizer, valve mounting provisions and their locations, a nozzle closure during 
storage, a thrust structure, and possibly an actuator and gimbal mount, if gim- 
baling is required by the mission. A detailed layout or CAD (computer-aided 
design) image (not shown in this analysis) would allow a more accurate picture 
and a good determination of the mass of the thrust chamber and its center of 
gravity both with and without propellants. 

Estimates of gas pressures, liquid pressures (or pressure drops) in the flow 
passages, injector, cooling jacket, and the valves are needed for the stress anal- 
ysis, so that various wall thicknesses and component masses can be determined. 
Material properties will need to be obtained from references or tests. A few of 
these analyses and designs may actually change some of the data we selected or 
estimated early in this sample analysis, and some of the calculated parameters 
may have to be re-analyzed and revised. Further changes in the thrust chamber 
design may become evident in the design of the engine, the tanks, or the inter- 
face with the vehicle. The methods, processes, and fixtures for manufacturing 
and testing (and the number and types of tests) will have to be evaluated and the 
number of thrust chambers to be built has to be decided before we can arrive at 
a reasonable manufacturing plan, a schedule, and cost estimates. 

PROBLEMS 

1. How much total heat per second can be absorbed in a thrust chamber with an inside 
wall surface area of 0.200 ra? if the coolant is liquid hydrogen and the coolant 
temperature does not exceed 145 K in the jacket? Coolant flow is 2 kg/sec. What is 
the average heat transfer rate per second per unit area? Use the data from Table 7-1 
and the following: 



2. During a static test a certain steel thrust chamber is cooled by water in its cooling 
jacket. The following data are given for the temperature range and pressure of the 
coolant: 



Heat of vaporization near boiling point 
Thermal conductivity (gas at 21 K) 



446 kJ/kg 



0.013 W/m-K 
0.128 W/m-K 
0.165 W/m-K 



(gas at 194.75 K) 
(gas at 273.15 K) 



Average water temperature 
Thermal conductivity of water 
Gas temperature 
Specific gravity of water 



100°F 

1.07 X 10-“* Btu/sec-ftT°F/ft 
4500°F 
1.00 



340 THRUST CHAMBERS 



Viscosity of water 
Specific heat of water 
Cooling passage dimensions 
Water flow through passage 
Thickness of inner wall 
Heat absorbed 

Thermal conductivity of wall 
material 



2.5 X 10“^ Ibf-sec/ft^ 
1.3 Btu/lb-°F 
i X i in. 

0.585 Ib/sec 
i in. 

1.3 Btu/in.^-sec 
26 Btu/hr-fP-°F/ft 



Determine (a) the film coefficient of the coolant; (b) the wall temperature on the 
coolant side; (c) the wall temperature on the gas side. 

3. In the example of Problem 2 determine the water flow required to decrease the wall 
temperature on the gas side by 100°F. What is the percentage increase in coolant 
velocity? Assume that the various properties of the water and the average water 
temperature do not change. 

4 . Determine the absolute and relative reduction in wall temperatures and heat transfer 
caused by applying insulation in a liquid-cooled rocket chamber with the following 
data: 



Tube wall thickness 
Gas temperature 
Gas-side wall temperature 
Heat transfer rate 
Liquid-film coefficient 
Wall material 



0.381 mm 
2760 K 
1260 K 

15 MW/m^-sec 
23 kW/m^-K 

Stainless steel AISI type 302 



A 0.2-mm-thick layer of insulating paint is applied on the gas side; the paint consists 
mostly of magnesia particles. The average conductivity of this magnesia paint is 
2.59W/m^-K/m over the temperature range. The stainless steel has an average thermal 
conductivity of 140 Btu/hr-ft^-°F/in. and a specific gravity of 7.98 

5. A small thruster has the following characteristics: 



Propellants 

Injection individual hole size 
Injection hole pattern 
Thrust chamber type 

Specific gravities 
Impingement point 
Direction of jet momentum 
r = 1.65 (fuel rich) 

F = 300 Ibf 
p\ = 250 psi 
(Ap)inj = 50.0 psi 



Nitrogen tetroxide and monomethyl 
hydrazine 

Between 0.063 and 0.030 in. 

Unlike impinging doublet 
Ablative liner with a carbon-carbon nozzle 
throat insert 

1.446 for oxidizer and 0.876 for fuel 

0.25 in. from injector face 

Parallel to chamber axis after impingement 

(?5)actual — 251 sec 

tt = 25 sec 

Ai/A, = 3.0 

{Cd)o = (Q)/ = 0.86 



PROBLEMS 341 



Determine the number of oxidizer and fuel injection holes and their angles. Make a 
sketch to show the symmetric hole pattern and the feed passages in the injector. To 
protect the wall, the outermost holes should all be fuel holes. 

6. A large, uncooled, uninsulated, low-carbon-steel thrust chamber burned out in the 
throat region during a test. The wall (0.375 in. thick) had melted and there were 
several holes. The test engineer said that he estimated the heat transfer to have 
been about 15 Btu/in.^. The chamber was repaired and you are responsible for the 
next test. Someone suggested that a series of water hoses be hooked up to spray 
plenty of water on the outside of the nozzle wall at the throat region during the 
next test to prolong the bring duration. The steel’s melting point is estimated to be 
2550°F. Because of the likely local variation in mixture ratio and possibly imperfect 
impingement, you anticipate some local gas regions that are oxidizer rich and could 
start the rapid oxidation of the steel. You therefore decide that 2150°F should be 
the maximum allowable inner wall temperature. Besides knowing the steel weight 
density (0.284 Ibf/in.^), you have the following data for steel for the temperature 
range from ambient to 2150°F: the specibc heat is 0.143 Btu/lbm-°F and the thermal 
conductivity is 260 Btu/hr-ft‘-°F/in. Determine the approximate time for running 
the next test (without burnout) both with and without the water sprays. Justify any 
assumptions you make about the liquid-blm coefficient of the water flow. If the 
water spray seems to be worth while (getting at least 10% more burning time), make 
sketches with notes on how the mechanic should arrange for this water flow so it will 
be most effective. 

7 . The following conditions are given for a double-walled cooling jacket of a rocket 
thrust chamber assembly: 



Rated chamber pressure 


210 psi 


Rated jacket pressure 


290 psi 


Chamber diameter 


16.5 in. 


Nozzle throat diameter 


5.0 in. 


Nozzle throat gas pressure 


112 psi 


Average inner wall temperature at throat region 


1100°F 


Average inner wall temperature at chamber region 


800°F 


Cooling passage height at chamber and nozzle exit 


0.375 in. 


Cooling passage height at nozzle throat 


0.250 in. 


Nozzle exit gas pressure 


14.7 psi. 


Nozzle exit diameter 


10 in. 


Wall material 


1020 carbon steel 


Inner wall thickness 


0.08 in. 


Safety factor on yield strength 


2.5 


Cooling fluid 


RP-1 


Average thermal conductivity of steel 


250 Btu/hr-ft^-°F/in. 



Assume other parameters, if needed. Compute the outside diameters and the thickness 
of the inner and outer walls at the chamber, at the throat, and at the nozzle exit. 

8. Determine the hole sizes and the angle setting for a multiple-hole, doublet impinging 
stream injector that uses alcohol and liquid oxygen as propellants. The resultant 



342 THRUST CHAMBERS 



momentum should be axial, and the angle between the oxygen and fuel jets (yo + yj) 
should be 60°. Assume the following: 



(Cd)„ 


0.87 


Chamber pressure 


300 psi 


(Cd)f 


0.91 


Fuel pressure 


400 psi 


Po 


71 Ib/ft^ 


Oxygen pressure 


380 psi 


Pf 


51 Ib/ft^ 


Number of jet pairs 


4 


r 


1.20 


Thrust 


250 Ibf 



Actual specihc impulse 218 sec 



Answers: 0.0197 in.; 0.0214 in.; 32.3°; 27.7°. 

9 . Table 11-6 shows that the RD-120 rocket engine can operate down to 85% of full 
thrust and at a mixture ratio variation of ±10.0%. In a particular static test the average 
thrust was held at 96% of nominal and the average mixture ratio was 2.0% fuel rich. 
Assume a 1.0% residual propellant, but neglect other propellant budget allowances. 
What percentage of the fuel and oxidizer that have been loaded will remain unused 
at thrust termination? If we want to correct the mixture ratio in the last 20.0% of the 
test duration and use up all the available propellant, what would be the mixture ratio 
and propellant flows for this last period? 

10 . Make a simple cross-section sketch approximately to scale of the thrust chamber that 
was analyzed in Section 8.8. The various dimensions should be close, but need not be 
accurate. Include or make separate detailed-section sketches of the cooling jacket and 
the injector. Also compile a table of all the key characteristics, similar to Table 8-1, 
but include gas generator flows and key materials. Make estimates or assumptions 
for any key data that is not mentioned in Section 8.8. 

SYMBOLS 

A area, rc? (ft^) 

Afl projected area of linear aerospike ramp, (ft^) 

Cp specific heat at constant pressure, J/kg-K (Btu/lbm°R) 

c average liquid specific heat, J/kg-K (Btu/lbm°R) 

Cci discharge coefficient 

Cp specific heat at constant pressure J/kg-K (Btu/lbm R) 

D diameter, m (ft) 

E modulus of elasticity, N/m^ (Ibf/in.^), or radiation energy, kg-m^/sec^ 

/ friction loss coefficient, or geometric factor in radiation or final 

(velocity) 

go sea-level acceleration of gravity, 9.806 m/sec^ (32.17 ft/sec^) 
h film coefficient, W/(m^-K); (Btu/(ft^-°R) 

Is specific impulse, sec 

k specific heat ratio 

L length, m (ft) 

L* characteristic chamber length, m (ft) 

m mass, kg 



SYMBOLS 343 



m mass flow rate, kg/sec (Ib/sec) 

p pressure, N/m^ or Pa (Ibf/in.^) 

Pr Prandtl number (Cp/r/A:) 

q heat transfer rate or heat flow per unit area, J/m^-sec (Btu/ft^-sec) 

Q volume flow rate, m^/sec (ft^/sec), or heat flow rate, J/sec 

R Reynold’s number Dvp/ p. 

r flow mixture ratio (oxidizer to fuel); or radius, m (ft) 

^ stress N/m^ (Ibf/in.^) 

t time, sec, or thickness, m (ft) 

ts stay time, sec 

tw wall thickness, m (in.) 

T absolute temperature, K (°R) 

Am flight velocity increment m/sec (If/sec) 

V velocity, m/sec (ft/sec) 

Vi specific volume, m^/kg (ft^/lb) 

Vc combustion chamber volume (volume up to throat), m^(ft^) 

X, y coordinates of a parabola with constant p 

Greek Letters 

Yo angle between chamber axis and oxidizer stream 

Yf angle between chamber axis and fuel stream 

A finite differential 

5 angle between chamber axis and the resultant stream 

e nozzle exit area ratio (e = A 2 IA,) 

e emmissivity of radiating surface, dimension gas less 

6 angle 

K thermal conductivity, J/(m^-sec-K)/m (Btu/in.^-sec^-°R/in.) 

X coefficient of thermal expansion, m/m-K (in./in. -°R) 

ji absolute gas viscosity, kg/(m-sec) or Ibf/ft^-sec 

V Poisson ratio 

p density, kg/m^(lbf/ft^) 

a Stefan-Boltzmann constant (5.67 x 10“* W/m^-K"^); also stress 

N/m2(lbf/in2) N/m2(lbf/in2) 



Subscripts 

am arithmetic mean 

c chamber 

/ fuel or final condition 

g gas 

gg gas generator 



344 



THRUST CHAMBERS 



inj 

1 

0 

1 

tc 

w 

wg 

wl 

0 

1 

2 
3 



injector 

liquid 

oxidizer 

throat 

thrust chamber 
wall 

wall on side of gas 

wall on side of liquid 

initial condition 

inlet or chamber condition 

nozzle exit condition 

atmosphere or ambient condition 



REFERENCES 

8-1. G. P. Sutton, History of Liquid Propellant Rocket Engines, AIAA, Reston, VA, 



8-2. V. Yang, M. Habiballah, J. Hulka, and M. Popp, (Eds.), Liquid Rocket Thrust 
Chambers: Aspects of Modeling, Analysis, and Design, Progress in Astronautics 
and Aeronautics (Series), Vol. 200, AIAA, Reston, VA, 2004. 

8-3. R. D. McKown, “Brazing the SSME,” Threshold, an Engineering Journal of Power 
Technology , No. 1, Rocketdyne Division of Rockwell International (now Pratt & 
Whitney Rocketdyne.), Canoga Park, CA, March 1987, pp. 8-13. 

8-4. R. A. Ellis, I. C. Lee, E. M. Payne, M. Lacoste, A. Lacombe, and P. loyes, 
“Testing of the RL lOB-2 Carbon-Carbon Nozzle Extension,” AIAA Conference 
Paper 98-3363, July 1998. 

8-5. M. Niino, A. Kumakawa, T. Hirano, K. Sumiyashi, and R. Watanabe, “Life Pre- 
diction of CIP Formed Thrust Chambers,” Acta Astronautica, Vol. 13, Nos. 6-7, 
1986, pp. 363-369 (fatigue life prediction). 

8-6. J. S. Porowski, W. J. O’Donnell, M. L. Badlani, B. Kasraie, and H. J. Kasper, 
“Simplified Design and Life Predictions of Rocket Thrustchambers,” Journal of 
Spacecraft and Rockets , Vol. 22, No. 2, March-April 1985, pp. 181-187. 

8-7. J. S. Kinkaid, “Aerospike Evolution,” Threshold, The Boeing Co., Rocketdyne 
Propulsion and Power, No. 18, Spring 2000, pp. 4-13. 

8-8. T. T. Bui, J. E. Murray, C. E. Rodgers, S. Bartel, A. Cesaroni, and M. Dennet, 
“Flight Research of an Aerospike Nozzle Using High Power Solid Rockets,” AIAA 
paper 2005-3797, 2005. 

8-9. A. J. Fortini and R. H. Tuffias, “Advanced Materials for Chemical Propulsion: 
Oxide-Iridium/Rhenium Combustion Chambers,” AIAA Paper 99-2894, June 1999. 

8-10. F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavine, Introduction to 
Heat Transfer, 5th d., John Wiley & Sons, Hoboken, NJ, 2006. 

8-11. A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer, Vol. 1. 
Mathematical Modeling and Vol. 2. The Finite Difference Methodology, John 
Wiley & Sons, New York, 1995 and 1996. 



2006. 



REFERENCES 345 



8-12. R. W. Lewis, Pemmal Nithiarasu, and Kankanhalli Seetharamu, Fundamentals of 
the Finite Element Method for Heat and Fluid Flow , John Wiley & Sons, Hoboken, 
NJ, 2004. 

8-13. D. R. Bartz, “Survey of Relationships between Theory and Experiment for Convec- 
tive Heat Transfer in Rocket Combustion Gases,” in Advances in Rocket Propul- 
sion, S. S. Penner (Ed.), AGARD, Technivision Services, Manchester, UK, 1968. 

8-14. E. Mayer, “Analysis of Pressure Feasibility Limits in Regenerative Cooling of 
Combustion Chambers for Large Thrust Rockets,” in Liquid Rockets and Propel- 
lants, L. E. Bollinger, M. Goldsmith, and A. W. Lemmon, Jr. (Eds.), Academic 
Press, New York, 1969, pp. 543-561. 

8-15. J. M. Fowler and C. F. Warner, “Measurements of the Heat-Transfer Coefficients 
for Hydrogen Flowing in a Heated Tube,” American Rocket Society Journal, Vol. 
30, No. 3, March 1960, pp. 266-267. 

8-16. P. A. Baudart, V. Duthoit, T. Delaporte, and E. Znaty, “Numerical Modeling of 
the HM 7 B Main Chamber Ignition,” AIAA Paper 89-2397, 1989. 

8-17. A. R. Casillas, J. Eninger, G. Josephs, J. Kenney, and M. Trinidad, “Control of 
Propellant Lead/Lag to the LEA in the AXAF Propulsion System,” AIAA Paper 
98-3204, July 1998. 



CHAPTER 9 



LIQUID PROPELLANT COMBUSTION 
AND ITS STABILITY 



In this chapter we treat the complex phenomena of the combustion processes in 
the combustion chamber of a liquid bipropellant thrust chamber. We describe 
in general terms the combustion behavior, the progress in analysis of combus- 
tion, the several types of combustion instability with its undesirable effects, and 
semiempirical remedies which avoid these. The objective is to operate at very 
high combustion efficiencies and to prevent 100% of all the occurrence of disrup- 
tive or destructive combustion instability. Thrust chambers should operate with 
stable combustion over their full range of operating conditions. For a treatment 
of these subjects see Refs. 9-1 to 9-7. 

The combustion of liquid propellants is very efficient in well-designed thrust 
chambers, precombustion chambers, or gas generators. Efficiencies of 95 to 
99.5% are typical compared to turbojets or furnaces, which can range from 
50 to 97%. This is due to the very high reaction rates at the high combustion 
temperatures and the thorough mixing of fuel and oxidizer reaction species by 
means of good injection flow distribution and gas turbulence. The losses are 
largely due to incomplete burning or inadequate mixing (nonuniform mixing 
ratio). For very small bipropellant thrust chambers or small gas generators, 
where the injector has very few injection orihces or elements, the combustion 
efficiency can be well below 95%. 



9.1. COMBUSTION PROCESS 

In describing the combustion processes, it is convenient and helpful to the 
understanding to divide the combustion chamber into a series of discrete zones, 
as shown in Fig. 9-1 for a typical configuration. It has a flat injector face 



346 



9.1 . COMBUSTION PROCESS 347 



Injection/ Rapid- Streamtube 

atomization combustion combustion Transonic-flow zone 




Supersonic 

expansion 

zone 



FIGURE 9-1. Division of combustion chamber into zones for analysis. (Modified from 
Y. M. Advanced Chemical Rocket Propulsion, Academic Press, New York, 1987.) 



with many small injection orifices for introducing both fuel and oxidizer liquids 
as many discrete individual streams, jets, or thin sprays or sheets. The relative 
thicknesses of these zones in the axial direction, their behavior, and their 
transitions are influenced by the specific propellant combination, the operating 
conditions (pressure, mixture ratio, etc.), the design of the injector, and chamber 
geometry. The boundaries between the zones shown in Fig. 9-1 are really not flat 
surfaces and do not display steady flow. They are undulating, dynamically mov- 
able, irregular boundaries with localized changes in velocity, temporary bulges, 
locally intense radiation emissions, or variable temperature. Table 9-1 shows 
the major interacting physical and chemical processes that occur in the chamber. 
This table is a modification of tables and data in Refs. 9-2, 9-3, and 9-7. 

The combustion behavior is propellant dependent. If the cryogenic fuel hydro- 
gen has been used to cool the thrust chamber, the hydrogen would be gaseous 
and fairly warm (60 to 240 K); there would be no liquid hydrogen droplets and 
no evaporation. With hypergolic propellants there is an initial chemical reaction 
in the liquid phase when a droplet of fuel impinges on a droplet of oxidizer. 
Experiments show that the contact can create local explosions and enough 
energy release to suddenly vaporize a thin layer of the fuel and the oxidizer 
locally at the droplet’s contact face; there immediately follows a vapor chemical 
reaction and a blow-apart and breakup of the droplets. Occasionally there are 
small local explosions and shock wave phenomena (Refs. 9-8 and 9-9). 

The combustion process is usually analyzed as a steady-flow process, but in 
reality it is not so. When observing any one location within the chamber, one 
finds a turbulent flow with local fluctuations with time in pressure, flow, tem- 
perature, density, mixture ratio, or radiation emissions. The boundaries between 
zones identified in Fig. 9-1 or Table 9-1 are not in a single plane, do not have 
steady flow, are more like an undulating changing surface, which locally moves 



348 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



TABLE 9-1. Physical and Chemical Processes in the Combustion of Liquid Propellants 
Injection Atomization Vaporization 



Liquid jets or sprays enter 
chamber at relatively low 
velocities 

Sometimes gas propellant is 
injected 

Partial evaporation of 
liquids 

Interaction of jets or sprays 
with high-pressure gas 



Impingement of jets or 
sheets 

Formation of liquid fans 
or spray cones 
Formation of droplets 
Secondary breakup of 
drops 

Liquid mixing and some 
liquid-liquid chemical 
reaction 

Oscillations of jets or fans 
or spray liquid sheets as 
they become unstable 
during breakup 
Vaporization begins and 
some vapor reactions 
occur 



Droplet gasification and 
diffusion 

Further heat release from 
local chemical reactions 
Relatively low gas velocities 
and some cross flow 
Heat absorbed by radiation, 
convection, and 
conduction, from blowback 
of turbulent gases from the 
hot reaction zone 
Acceleration to higher 
velocities 

Vaporization rate influenced 
by turbulence, pressure or 
temperature oscillations 
and acoustic waves 



Mixing and Reaction Expansion in Chamber 



Turbulent mixing (three-dimensional) 
Multiple chemical reactions and major 
heat releases 

Interactions of turbulence with droplets 
and chemical reactions 
Temperature rise reduces densities and 
increases volume flow 
Local mixture ratios, reaction rates, or 
velocities are not uniform across 
chamber and vary rapidly with time 
Some tangential and radial flows, much of 
it near the injector 



Chemical kinetics causes attainment of 
final combustion femperafure and final 
equilibrium reacfion gas composifion 
Gas dynamics displays turbulence and 
increasing axial gas velocities 
Formation of a boundary layer 
Acceleration to high chamber velocities 
Streamlined high-velocity axial flow 
with very little cross flow 
Flow toward nozzle 
Heat transfer to walls 



forward and backward. Some of the processes listed in the table occur sequen- 
tially, while others occur simultaneously. Not all the listed processes happen 
with each propellant. Three-dimensional turbulence happens in all parts of the 
combustion chamber and nozzle. Most processes also create noise. 

Rapid Combustion Zone 

In this zone intensive and rapid chemical reactions occur at increasingly 
higher temperature; any remaining liquid droplets are vaporized by convective 
heating and gas pockets of fuel-rich and fuel-lean gases are mixed. The mixing 
is accomplished by local three-dimensional turbulence and diffusion of the 
gas species. 



9.1 . COMBUSTION PROCESS 349 



The further breakdown of the propellant chemicals into intermediate fractions 
and smaller, simpler chemicals and the oxidation of fuel fractions occur rapidly 
in this zone. The rate of heat release increases greatly, and this causes the specific 
volume of the gas mixture to increase and the local axial velocity to increase by 
a factor of 100 or more. The rapid expansion of the heated gases also forces a 
series of small local transverse gas flows (cross flow) and even some localized 
temporary back flow from hot high-buming-rate sites to colder low-burning- 
rate sites. The liquid droplets that may still persist in the upstream portion of 
this zone, do not follow the gas flow quickly, and are difficult to move in a 
transverse direction. Therefore, zones of fuel-rich or oxidizer-rich gases will 
persist according to the orifice spray pattern in the upstream injection zone. 
The gas composition and mixture ratio across the chamber section become more 
uniform as the gases move through this zone, but the mixture never becomes 
truly uniform. As the reaction product gases are accelerated, they become hotter 
(due to further heat releases) and the lateral velocities remain relatively small 
compared to the increasing axial velocities. Our understanding of the process is 
aided by high-speed photography See Ref. 9-7. 



Injection/Atomization Zone 

Two different liquids are injected, both with storable bipropellants and/or liquid 
oxygen-hydrocarbon combinations. They are injected through orifices or slots at 
velocities typically between 7 and 60 m/sec or about 20 to 200 ft/sec. The injector 
design has a profound influence on the combustion behavior and some seemingly 
minor design changes can have a major effect on instability. The pattern, sizes, 
number, distribution, and types of orifices influence the combustion behavior, as 
do the pressure drop, manifold geometry, or surface roughness in the injection 
orifice walls. The individual jets, streams, or sheets break up into droplets by 
impingement of one jet with another (or with a surface), by impingement of 
conical spray sheets, by the inherent instabilities of liquid jets or sprays, or by 
the interaction with gases at a different velocity and temperature. In this first 
zone the liquids are atomized into a large number of small droplets (see Refs. 
9-3, 9-6, 9-7, 9-9, and 9-10). Heat is transferred to the droplets by radiation 
from the very hot rapid combustion zone and by convection from moderately hot 
gases in the first zone. The droplets evaporate and create local regions rich either 
in fuel vapor or oxidizer vapor. 

This first zone is heterogeneous; it contains liquids and vaporized propellant 
as well as some burning hot gases. With the liquid being located at discrete sites, 
there are large gradients in all directions with respect to fuel and oxidizer mass 
fluxes, mixture ratio, size and dispersion of droplets, or properties of the gaseous 
medium. Chemical reactions begin to occur in this zone, but the rate of heat 
generation is relatively low, in part because the liquids and the gases are still 
relatively cold and in part because vaporization near the droplets causes fuel-rich 
and fuel-lean regions which do not bum as quickly. Some hot gases from the 
combustion zone are recirculated back from the rapid combustion zone, and they 



350 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



can create local gas velocities that flow across the injector face. The hot gases, 
which can flow in unsteady vortexes or turbulence patterns, are essential to the 
initial evaporation of the liquids. 

The injection, atomization, and vaporization processes are different if one of 
the propellants is a gas. For example, this occurs in liquid oxygen with gaseous 
hydrogen propellant in thrust chambers or precombustion chambers, where 
liquid hydrogen has absorbed heat from cooling jackets and has been gasified. 
Hydrogen gas has no droplets and does not evaporate. The gas usually has a 
much higher injection velocity (above 120 m/sec) than the liquid propellant. 
This causes shear forces to be imposed on the liquid jets, with more rapid 
droplet formation and gasification. The preferred injector design for gaseous 
hydrogen and liquid oxygen is different from the individual jet streams used 
with storable propellants, as shown in Chapter 8. 



Stream Tube Combustion Zone 

In this zone oxidation reactions continue, but at a lower rate, and some addi- 
tional heat is released. However, chemical reactions continue because the mixture 
tends to be driven toward an equilibrium composition. Axial velocities are high 
(200 to 600 m/sec), and the transverse convective flow velocities are relatively 
small. Streamlines are formed and there is relatively little turbulent mixing across 
streamline boundaries. Locally the flow velocity and the pressure fluctuate some- 
what. The residence time in this zone is very short compared to the residence 
time in the other two zones. The streamline type, inviscid flow, and the chemical 
reactions toward achieving chemical equilibrium presist not only throughout the 
remainder of the combustion chamber, but are also extended into the nozzle. 

The residence time of the propellant material in the combustion chamber is 
very short, usually less than 10 msec. Combustion in a liquid rocket engine is very 
dynamic, with the volumetric heat release being approximately 370 MJ/m^-sec, 
which is much higher than in turbojets. Further, the higher temperature in a rocket 
causes chemical reaction rates to be several times faster (increasing exponentially 
with temperature) than in turbojet. 



9.2. ANALYSIS AND SIMULATION 

For the purpose of analyzing the combustion process and its instabilities, it 
has been convenient to divide the acoustical characteristics into linear and 
nonlinear behavior. A number of computer simulations with linear analyses 
have been developed over the last 45 years and have been used to understand 
the combustion process with liquid propellant combustion devices and to predict 
combustion oscillation frequencies. The nonlinear behavior (e.g., why does a dis- 
turbance cause an apparently stable combustion to suddenly become unstable?) 
is not well understood and not properly simulated. Mathematical simulations 
require a number of assumptions and simplifications to permit feasible solutions 



9.2. ANALYSIS AND SIMULATION 351 



(see Refs. 9-1 to 9-5, 9-7, 9-11, and 9-12). Good models exist for relatively 
simple phenomena such as droplets of a propellant vaporizing (see Refs. 9-8 
and 9-10) and burning in a gaseous atmosphere or the steady-state flow of gases 
with heat release from chemical reactions. The thermochemical equilibrium 
principles mentioned in Chapter 5 also apply here. Some programs consider 
turbulence and film cooling effects. 

The following phenomena are usually ignored or greatly simplified: cross 
flows; nonsymmetrical gradients; unsteadiness of the flow; time variations in the 
local temperature, local velocity, or local gas composition; thermochemical reac- 
tions at local off-design mixture ratios and at different kinetic rates; enhancement 
of vaporization by acoustic fields (see Ref. 9-9); uncertainties in the spatial as 
well as the size distribution of droplets from sprays; or drag forces on droplets. 
It requires skilled, experienced personnel to use, interpret, and modify the more 
complex programs so that meaningful results and conclusions can be obtained. 
The outputs of these computer programs can give valuable help and confirmation 
about the particular design and are useful guides in interpreting actual test results, 
but by themselves they are not sufficient to determine the designs, select specific 
injector patterns, or predict the occurrence of combustion instabilities. 

All the existing computer programs known to the authors are suitable for 
steady-state flow conditions, usually at a predetermined average mixture ratio and 
chamber pressure. However, during the starting, thrust change (such as throttling), 
and stopping transients, the mixture ratio and the pressure can change drastically. 
The analysis of these transient conditions is more difficult. 

The combustion is strongly influenced by the injector design. The following 
are some of the injection parameters which influence combustion behavior: 
injector spray or jet pattern, their impingement, hole sizes or hole distribu- 
tion, droplet evaporation, injection pressure drop, mixture ratio, pressure or 
temperature gradients near the injector, chamber/injector geometry, pressurant 
gas saturation, initial propellant temperature, and liquid injection pressure drop. 
Attempts to analyze these effects have met with only partial success. 

Computational fluid dynamics (CFD) is a relatively new analytical tool 
that can provide a comprehensive description of complex fluid dynamic and 
thermodynamic behavior. It allows for a time history of all parameters and can 
even include some nonlinear effects. Numerical approaches are used to evaluate 
sets of equations and models that represent the behavior of the fluid. For 
complex geometries the information has been tracked with more than 150,000 
discrete locations and can include changes in gas composition, thermodynamic 
conditions, equilibrium reactions, phase changes, viscous or nonviscous flow, 
one-, two-, or three-dimensional flow and steady-state or transient conditions. It 
has been applied to resonance cavities in injectors or chambers and to the flow 
of burning gases through turbines. A comprehensive rocket combustion model 
using CFD is not yet available, but could become useful in the future. 



352 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



9.3. COMBUSTION INSTABILITY 

If the process of rocket combustion is not controlled (by proper design), then 
combustion instabilities can occur which can very quickly cause excessive pres- 
sure vibration forces (which may break engine parts) or excessive heat transfer 
(which may melt thrust chamber parts). The aim is to prevent all occurrences 
of this instability and to maintain reliable operation (see Refs. 9-5 and 9-11 to 
9-14). Much progress has been made in understanding and avoiding combustion 
instability. 

References 9-1 to 9-3 and 9-5 to 9-7 describe the analysis and behavior 
of combustion instability in thrust chamberts as practiced in the United States. 
References 9-13 and 9-14 present the same but as practiced in Russia and its 
predecessor the Soviet Union. There are some differences in the assumptions, 
analysis, and testing, but the basic approach is essentially similar. In both coun- 
tries any new or modified thrust chamber and/or rocket engine has to be tested 
to demonstrate that combustion instability will not occur. 

Table 9-2 lists the principal types of combustion vibrations encountered in 
liquid rocket thrust chambers (see Refs. 9-3 and 9-9). Admittedly, combustion 
in a liquid rocket is never perfectly smooth; some fluctuations of pressure, tem- 
perature, and velocity are always present. When these fluctuations interact with 
the natural frequencies of the propellant feed system (with and without vehicle 
structure) or the chamber acoustics, periodic superimposed oscillations, recog- 
nized as instability, occur. In normal rocket practice smooth combustion occurs 
when pressure fluctuations during steady operation do not exceed about ±5% of 
the mean chamber pressure. Combustion that gives greater pressure fluctuations 
at a chamber wall location which occur at completely random intervals is called 
rough combustion. Unstable combustion, or combustion instability, displays orga- 
nized oscillations occurring at well-defined intervals with a pressure peak that 
may be maintained, may increase, or may die out. These periodic peaks, repre- 
senting fairly large concentrations of vibratory energy, can be easily recognized 
against the random-noise background (see Fig. 9-2). 

Chugging, the first type of combustion instability listed in Table 9-2, stems 
mostly from the elastic nature of the feed systems and structures of vehicles or 
the imposition of propulsion forces upon the vehicle. Chugging of an engine 
or thrust chamber assembly can occur in a test facility, especially with low cham- 
ber pressure engines (100 to 500 psia), because of propellant pump cavitation, 
gas entrapment in propellant flow, tank pressurization control fluctuations, and 
vibration of engine supports and propellant lines. It can be caused by resonances 
in the engine feed system (such as an oscillating bellows inducing a periodic 
flow fluctuation) or a coupling of structural and feed system frequencies. 

When both the vehicle structure and the propellant liquid in the feed system 
have about the same natural frequency, then force coupling can occur, not 
only to maintain but also to strongly amplify oscillations. Propellant flow rate 
disturbances, usually at 10 to 50 Hz, give rise to low-frequency longitudinal 
combustion instability, producing a longitudinal motion of vibration in the 



9.3. COMBUSTION INSTABILITY 353 



TABLE 9-2. Principal Types of Combustion Instability 



Type and Word Description 


Frequency 
Range (Hz) 


Cause Relationship 


Low frequency, called chugging 
or feed system instability 


10-400 


Linked with pressure interactions 
between propellant feed system, 
if not the entire vehicle, and 
combustion chamber 


Intermediate frequency, called 
acoustic,® buzzing, or entropy 
waves 


400-1000 


Linked with mechanical vibrations 
of propulsion structure, injector 
manifold, flow eddies, 
fuel/oxidizer ratio fluctuations, 
and propellant feed system 
resonances 


High frequency, called screaming, 
screeching, or squealing 


Above 1000 


Linked with combustion process 
forces (pressure waves) and 
chamber acoustical resonance 
properties 



“Use of the word acoustic stems from the fact the frequency of the oscillations is related to com- 
bustion chamber dimensions and velocity of sound in the combustion gas. 



vehicle. This vehicle flight instability phenomenon has been called pogo 
instability since it is similar to pogo jumping stick motion. Pogo instabilities 
can occur in the large, long propellant feed lines of large vehcles such as space 
launch vehicles or ballistic missiles. See Refs. 9-15 to 9-17. 

Avoiding objectionable engine-vehicle coupled oscillation is best accom- 
plished at the time of initial design of the vehicle, as contrasted to applying 
“fixes” later as has been the case with rocket engines for the Thor, Atlas, and 
Titan vehicles. Analytical methods exist for understanding the vibration modes 
and damping tendencies of major vehicle components, including the propellant 
tanks, tank pressurization systems, propellant flow lines, engines, and basic vehi- 
cle structure. Figure 9-3, a simplified spring-mass model of a typical two-stage 
vehicle, indicates the complexity of the analytical problem. Fortunately, the vibra- 
tional characteristics of the assembly can be affected substantially by designing 
damping (energy absorption) into the major components or subassemblies. Tech- 
niques for damping pogo instability include the use of energy absorption devices 
in fluid flow lines, perforated tank liners, special tank supports, and properly 
designed engine, interstage, and payload support structures. 

A partially gas-filled pogo accumulator has been an effective damping device; 
it is attached to the main propellant feed line. Such an accumulator is used in 
the oxidizer feed line of the Space Shuttle Main Engine (SSME) betwen the two 
oxidizer turbopumps; it can be seen in Eigs. 6-1 and 6-12. The SSME fuel line 
does not need such a damping device because the fuel has a relatively very low 
density and a lower mass flow. 



354 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



Smooth combustion 




Time 





FIGURE 9-2. Typical oscillograph traces of chamber pressure p\ with time for different 
combustion events. 

The dynamic characteristics of a propellant pump can also have an influence 
on the pogo-type vibrations, as examined in Ref. 9-17. The pogo frequency will 
change as propellant is consumed and the remaining mass of propellant in the 
vehicle changes. The bending or flexing of pipes, joints or bellows, or long tanks 
also has an influence. 

Buzzing, the intermediate type of instability, seldom represents pressure 
perturbations greater than 5% of the mean in the combustion chamber and 
usually is not accompanied by large vibratory energy. It often is more noisy 
and annoying than damaging, although the occurrence of buzzing may initiate 
high-frequency instability. Often it is characteristic of coupling between the 
combustion process and flow in a portion of the propellant feed system. 
Initiation is thought to be from the combustion process. Acoustic resonance of 
the combustion chamber with a critical portion of the propellant flow system, 
sometimes originating in a pump, promotes continuation of the phenomenon. 
This type of instability seems to be more prevalent in medium-size engines 
(2000 to 250,000 N thrust or about 500 to 60,000 Ibf) than in large engines. 



9.3. COMBUSTION INSTABILITY 



355 













mi -Payload 



mj-Spacer, dome, 
5^ barrel 



ms-Oxidizer 

m^-Dome, ^ barrel 
m5- Between tanks 
ms- Dome, barrel 
m^-Fuei 

mg- Dome, barrel 
mg-Tail skirt 

mi(j-Engine 
mil -Forward skirt 
mi2-Dome, H barrel 

mi3- Oxidizer 



m,4-Dome, \ barrel 
between tanks 
mij-Dome, barrel 
^ between tanks 



mis-Fuel 



miy-Fuel cone, 
barrel 

mig-Pumps 
mig-Tail skirt 



A A 

/ \ m,o- Engines 



FIGURE 9-3. Typical two-stage vehicle spring-mass model used in analysis of pogo 
vibration in the vertical direction. 



356 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



The third type, screeching or screaming, has high frequency and is most 
perplexing and most common in the development of new engines. Both liquid and 
solid propellant rockets commonly experience high-frequency instability during 
their development phase. Since energy content increases with frequency, this 
type is the most destructive, capable of destroying an engine in much less than 
1 sec. Once encountered, it is the type for which it is most difficult to prove that 
the incorporated “fixes” or improvemenfs render the engine “stable” under all 
launch and flight conditions. It can be treated as a phenomenon isolated to the 
combustion chamber and not generally influenced by feed system or structure. 

High-frequency instability occurs in at least two modes, longitudinal and 
transverse. The longitudinal mode (sometimes called organ pipe mode) prop- 
agates along axial planes of the combustion chamber, and the pressure waves 
are reflected at the injector face and the converging nozzle cone. The trans- 
verse modes propagate along planes perpendicular to the chamber axis and can 
be broken down into tangential and radial modes. Transverse mode instability 
predominates in large liquid rockets, particularly in the vicinity of the injec- 
tor. Figure 9-4 shows the distribution of pressure at various time intervals in 
a cylindrical combustion chamber (cross section) encountering transverse mode 
instability. Two kinds of wave form have been observed for tangential vibrations. 
One can be considered a standing wave that remains fixed in position while its 
pressure amplitude fluctuates. The second is a spinning or traveling tangential 
wave which has associated with it a rotation of the whole vibratory system. This 
waveform can be visualized as one in which the amplitude remains constant 
while the wave rotates. Combinations of transverse and longitudinal modes can 
also occur and their frequency can also be estimated. See Refs. 9-3 to 9-6. 

Energy that drives screeching is believed to be predominantly from 
acoustically stimulated variations in droplet vaporization and/or mixing, local det- 
onations, and acoustic changes in combustion rates. Thus, with favorable acoustic 
properties, high-frequency combustion instability, once triggered, can rapidly 
drive itself into a destructive mode. Invariable, a distinct boundary layer seems 
to disappear and heat transfer rates increase by an order of magnitude, much 
as with detonation, causing metal melting and wall bum-throughs, sometimes 
within less than 1 sec. The tangential modes appear to be the most damaging, 
heat transfer rates during instability often increasing 4 to 10 times. Often the 
instantaneous pressure peaks are about twice as high as with stable operation. 

One possible source of triggering high-frequency instability is a rocket 
combustion phenomenon called popping. Popping is an undesirable random 
high-amplitude pressure disturbance that occurs during steady-state operation of 
a rocket engine with hypergolic propellants. It is a possible source for initiation 
of high-frequency instability. “Pops” exhibit some of the characteristics of a 
detonation wave. The rise time of the pressure is a few microseconds and the 
pressure ratio across the wave can be as high as 7:1. The elimination of popping 
is usually achieved by redesign of the injector rather than by the application of 
baffles or absorbers. They are explained later in this chapter. 



9.3. COMBUSTION INSTABILITY 357 






N 




(Initial condition) 




(Half cycle later) 




FIGURE 9-4. Simplified representation of transverse pressure oscillation modes at two 
time intervals in a cylindrical combustion chamber. The solid line curves indicate pressures 
greater than the normal or mean operating pressure and the dashed lines indicate lower 
pressures. The N-N lines show the node locations for these wave modes. 



Some combustion instabilities can be induced by pulsations in the liquid flow 
originating in turbopumps. Unsteady liquid flow can be caused by irregular cavi- 
tation at the leading edge of the inducer impellers or of the main pump impellers 
(Ref. 9-18). Also, when an impeller’s trailing edge passes a rib or stationary vane 
of the volute, a small pressure perturbation always occurs in the liquid flow that 
travels downstream to the injector. These two types of pressure fluctuation can 
be greatly amplified if they coincide with the natural frequency of combustion 
vibrations in the chamber. 



358 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



The estimated natural frequencies can be determined from the wavelength /, 
or the distance traveled per cycle, and the acoustic velocity a (see Eq. 3-10). 
The frequency, or number of cycles per second, is 



Frequency = a/l = {\/l)^kTR' /^K (9-1) 

where k is the specific heat ratio, R' the universal gas constant, DJI the estimated 
molecular weight of the hot chamber gases, and T the local average absolute 
temperature. The length of wave travel depends on the vibrational mode, as 
shown in Fig. 9-4. Smaller chambers give higher frequencies. 

Table 9-3 shows a list of estimated vibration frequencies for one version of 
the Vulcain HM 60 rocket thrust chamber; it operates with liquid hydrogen and 
liquid oxygen propellants at a vacuum thrust of 1008 kN, a nominal chamber 
pressure of 10 MPa, and a nominal mixture ratio of 5.6 (see Ref. 9-19). The 
data in the table are based on acoustic measurements at ambient conditions with 
corrections for an appropriate sonic velocity correlation; since the chamber has 
a shallow conical shape and no discrete converging nozzle section, the purely 
longitudinal vibration modes would be weak; in fact, no pure longitudinal modes 
were detected. 

Figure 9-5 shows a series of time-sequenced diagrams of frequency- 
pressure-amplitude measurements taken in the oxygen injector manifold of 
the Vulcain HM 60 engine during the first 8 sec of a static thrust chamber 
test while operating at off-nominal design conditions. Chugging can be seen 
at low frequency (up to 500 Hz) during the first few seconds, and a natural 
frequency around 1500 Hz is attributed to the natural resonance frequency of the 
oxygen injector dome structure where the high-frequency pressure transducer 
was mounted. The continued oscillations observed at about 500 and 600 Hz are 
probably resonances associated with the feed system. 



TABLE 9-3. Estimated Acoustic Hot Gas Frequencies for Nominal Chamber 
Operating Conditions for the Vulcain HM-60 Thrust Chamber 



Mode" 


(L, T, R) 


Frequency (Hz) 


Mode" 


(L, T, R) 


Frequency (Hz) 


T1 


(0, 1, 0) 


2424 


L1T3 


(1, 3, 0) 


6303 


LlTl 


(1, 1,0) 


3579 


T4 


(0, 4, 0) 


6719 


T2 


(0, 2, 0) 


3856 


L2R1 


(2, 0, 1) 


7088 


R1 


(0, 0, 1) 


4849 


T5 


(0, 5, 0) 


8035 


L1T2 


(1, 2, 0) 


4987 


TR21 


(0, 2, 1) 


8335 


T3 


(0, 3, 0) 


5264 


R2 


(0, 0, 2) 


8774 


LlRl 


(1,0, 1) 


5934 









Reprinted with AIAA permission from Ref. 9-19. 

"Modes are classified as L (longitudinal), T (tangential), or R (radial), and the number refers to the 
first, second, or third natural frequency. 



9.3. COMBUSTION INSTABILITY 359 




Frequency (Hz) 



FIGURE 9-5. Graphical representation of a series of 40 superimposed frequency- 
amplitude diagrams taken 0.200 sec apart during the start phase (for the first 8 sec) of 
the Vulcain HM 60 thrust chamber. In this static hot-firing test the thrust chamber was 
operating at 109 bar chamber pressure and an oxidizer-to-fuel mass flow mixture ratio of 
6.6. (Used with permission from Ref. 9-19.) 



Rating Techniques 

Semiempirical techniques exist for artificially disturbing combustion in a rocket 
thrust chamber during test operation and evaluating its resistance to instability 
(see Refs. 9-6 and 9-20). These include: (1) nondirectional “bombs” placed 
within the combustion chamber; (2) oriented explosive pulses from a “pulse 
gun” directed through the chamber sidewall; and (3) directed flows of inert 
gas through the sidewall into the chamber. Often heavy prototype thrust 
chambers are used because they are less expensive and more resistant to damage 
than flight-weight engines. Other techniques used less widely but which are 
important, especially for small engines include: (1) momentary operation at 
“off-mixture ratio;” (2) introduction of “slugs” of inert gas into a propellant 
line; and (3) a purposeful “hard start” achieved by introducing a quantity of 
unreacted propellant at the beginning of the operation. 

The objective of these rating techniques is to measure and demonstrate the 
ability of an engine system to return quickly to normal or rated operation and 
stable combustion after the combustion process has intentionally been disturbed 
or perturbed. 

All techniques are intended to introduce shock waves into the combustion 
chamber or to otherwise perturb the combustion process, affording opportunity 
for measuring recovery time from a predetermined overpressure disturbance, 
assuming stable combustion resumes. Important to the magnitude and 
mode of the instability are the type of explosive charge selected, the size of 
the charge, the location and direction of the charge, and the duration of the 
exciting pulse. The bottom curve in Fig. 9-2 characterizes the recovery of 
stable operation after a combustion chamber was “bombed.” The time interval 
to recover and the magnitude of explosive or perturbation pressure are then 
used to rate the resistance of the engine to instability. 



360 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



The nondirectional bomb method and the explosive pulse-gun method are 
the two techniques in more common use. The bomb that can be used in large 
flight-weight thrust chambers without modification consists of six charges of 
250 grains or 16.2 grams of explosive (PETN, RDX, or similar) encased in 
a Teflon, nylon, or micarta case. Detonation of the bomb is achieved either 
electrically or thermally. Although the pulse gun requires modihcation of 
a combustion chamber, this technique affords directional control, which is 
important to tangential modes of high-frequency instability and allows several 
data points to be observed in a single test run by installing several pulse guns 
on one combustion chamber. Charges most frequently used are 10, 15, 20, 40, 
and 80 grains of pistol powder. Pulse guns can be fired in sequence, introducing 
successive pressure perturbations (approximately 150 msec apart), each of 
increasing intensity, into the combustion chamber. 



Control of Instabilities 

The control or really the complete elimination of instabilities is an important 
task during the design and development of a rocket engine. The designer usually 
relies on prior experience with similar engines and tests on new experimental 
engines. Analytical tools are available with which to simulate and evaluate the 
combustion process. The design selection has to be proven in actual experiments 
to be free of instabilities over a wide range of transient and steady-state operating 
conditions. Some of the experiments can be accomplished on a subscale rocket 
thrust chamber that has a similar injector, but most tests have to be done on a 
full-scale engine. 

The design features to control instabilities are different for the three types 
of vibrations described in Table 9-2. Chugging is usually avoided if there is no 
resonance in the propellant feed system or its coupling with the elastic vehicle 
structure. Increased injection pressure drop and the addition of artihcial damping 
devices in the propellant feed lines have been used successfully. Chugging and 
acoustical instabilities sometimes relate to the natural frequency of a particular 
feed system component that is free to oscillate, such as a loop of piping that can 
vibrate an injector dome, or a bellows whose oscillations cause a pumping effect. 

With the choice of the propellant combination usually fixed early in the plan- 
ning of a new engine, the designer can alter combustion feedback (depressing the 
driving mechanism) by altering injector details (such as changing the injector hole 
pattern, hole sizes, or by increasing the injection pressure drop), or alternatively 
by increasing acoustical damping within the combustion chamber. Of the two 
methods, the second has been favored in recent years because it is very effective, 
it is better understood, and theory fits. This leads to the application of injector 
face baffles, discrete acoustic energy absorption cavities, and combustion chamber 
liners or changes in injector design, often by using a trial-and-error approach. 

Injector face baffles (see Fig. 9-6) were a widely accepted design practice in 
the 1960s for overcoming or preventing high-frequency instability. Baffle design 
is predicated on the assumption that the most severe instability oscillations, along 



9.3. COMBUSTION INSTABILITY 361 




Ignition flame tube 



Oxygen 
from 
oxygen valve 



Spark 



Fluted oxidUer posts where 
hot hydrogen evaporates the oxygen 



Fuel inlet from 
hot gas manifold 

Cold hydrogen cavity 



Five compartment 
baffle with 75 cooled 
injection posts 



Primary injection 
plate (transpiration 
cooled) with 525 main 
injection elements 



Oxygen inlet 
manifolds 



Thrust 

load 

transmitting 
cone 



FIGURE 9-6. Main injector assembly of the Space Shuttle Main Engine showing baffle 
with five outer compartments. (Courtesy of Pratt & Whitney Rocketdyne.) 



with the driving source, are located in or near the injector-atomization zone at the 
injector end of the combustion chamber. The baffles minimize influential coupling 
and amplification of gas dynamic forces within the chamber. Obviously, baffles 
must be strong, have excellent resistance to combustion temperatures (they are 
usually cooled by propellant), and must protrude into the chamber enough to be 
effective, yet not so far as to act like an individual combustion chamber with 
its own acoustical characteristics. The number of baffle compartments is usually 
odd. An even number of compartments enhances the standing modes of insta- 
bility, with the baffles acting as nodal lines separating regions of relatively high 
and low pressure. The design and development of baffles remains largely empir- 
ical. Generally, baffles are designed to minimize acoustical frequencies below 
4000 Hz, since experience has shown damaging instability is rare at frequencies 
above 4000 Hz. 

Various mechanisms of energy absorption or vibration damping exist in a 
thrust chamber. Damping by wall friction in combustion chambers is not sig- 
nificant. The exhaust nozzle produces the main damping of longitudinal mode 
oscillations; the reflection of waves from the convergent nozzle entrance departs 
from that of an ideal closed end. The principal damping source affecting propa- 
gation in the transverse plane is combustion itself. The great volumetric change 
in going from liquid to burned gases and the momentum imparted to a particle 
(solid or liquid) both constitute damping phenomena in that they take energy 
from high instantaneous local pressures. Unfortunately, the combustion process 
can generate a great deal more pressure oscillation energy than is absorbed by 
its inherent damping mechanism. 



362 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 






Volume ^ Orifice 

V , ' diameter 

T d 



// y/. 



L + AL 
Helmholtz resonator 



Spring 

1 Mass 



y//////////////M 

Friction resistance 

Mechanical analogy 
of acoustic cavity 



Chamber \wall 




two cavities cavities 

FIGURE 9-7. Diagram of acoustic energy absorber cavities at the periphery of an injec- 
tor. In this thrust chamber the cavity restriction is a slot (in the shape of sections of a 
circular arc) and not a hole. Details of the chamber cooling channels, injector holes, or 
internal feed passages are not shown. 



Acoustical absorbers are applied usually as discrete cavities along or in the 
wall of the combustion chamber near the injector end. Both act as a series of 
Helmholtz resonators that remove energy from the vibratory system which other- 
wise would maintain the pressure oscillations. Figure 9-7 shows the application 
of discrete cavities (interrupted slots) at the “comer” of the injector face. The cor- 
ner location usually minimizes the fabrication problems, and it is the one location 
in a combustion chamber where a pressure antinode exists for all resonant modes 
of vibration, including longitudinal, tangential, radial, and combinations of these. 
Velocity oscillations are minimal at this point, which favors absorber effective- 
ness. Transverse modes of instability are best damped by locating absorbers at 
the corner location. Figure 9-7 also shows a Helmholtz resonator cavity and its 
working principles in simple form. Taking one resonator element, the mass of 
gas in the orifice with the volume of gas behind it forms an oscillatory system 
analogous to the spring-mass system shown (see Ref. 9-21). 

Absorption cavities designed as Helmholtz resonators placed in or near the 
injector face offer relatively high absorption bandwidth and energy absorbed 
per cycle. The Helmholtz resonator (an enclosed cavity with a small passage 
entry) dissipates energy twice each cycle (jets are formed upon inflow and out- 
flow). Modern design practice favors acoustic absorbers over baffles. The storable 



9.3. COMBUSTION INSTABILITY 363 



propellant rocket engine shown in Fig. 8-2 has acoustic absorption cavities in 
the chamber wall at a location next to the injector. 

The resonance frequency / of a Helmholtz cavity can be estimated as 



Here a is the local acoustic velocity, A is the restrictor area, A = {7r/4)d^, and 
other symbols are as shown in Fig. 9-7. The AL is an empirical factor between 
0.05 and 0.9L to allow for additional oscillating gas mass. It varies with the 
L/d ratio and the edge condition of the restricted orifice (sharp edge, rounded, 
chamfered). Resonators in thrust chambers are tuned or designed to perform their 
maximum damping at predicted frequencies. 

Small changes in injector geometry or design can cause an unstable combustion 
to become stable and vice versa. New injectors, therefore, use the design and 
geometry of proven, stable prior designs with the same propellants. For example, 
the individual pattern of concentric tube injector elements used with gaseous 
hydrogen and liquid oxygen (shown in Fig. 8-4d) are likely to be more stable, 
if the hydrogen gas is relatively warm and the injection velocity of the hydrogen 
is at least 10 times larger than that of the liquid oxygen. 

There are other combustion vibration remedies besides those that have 
already been mentioned, such a baffles or resonance cavities. This includes 
higher injection pressure drop, avoiding resonance with certain key engine 
component structures, changing critical injector dimension details, changing or 
modifying the propellant, or adding liners. References 9-3 and 9-6 describe 
several others, such as cooled flame holders, eliminating the entry of gases 
into the chamber during combustion, changing the locations of maximum heat 
release, varying critical dimensions on the injector elements, varying the number 
of injector elements and their distribution over the cross section of the chamber, 
changing the mixture ratio to be nonuniform near the face of the injector, or 
temporary consumable baffles, which are effective only during the start. 

In summary, the designer needs to (I) use data from prior successful engines 
and simulation programs to establish key design features and estimate the likely 
resonances of the engine and its major components, (2) design the feed system 
and structure to avoid these resonances, (3) use a robust injector design that will 
provide good mixing and dispersion of propellants and be resistant to distur- 
bances, and (4) if needed, include tuned damping devices (cavities) to overcome 
acoustic oscillations. To validate that a particular thrust chamber is stable, it is 
necessary to test it over the range of likely operating conditions without encoun- 
tering instability. An analysis is needed to determine the maximum and minimum 
likely propellant temperatures, maximum and minimum probable chamber pres- 
sures, and the highest and lowest mixture ratios, using a propellant budget as 
shown in Section 11.1. These limits then establish the variations of test condi- 
tions for this test series. Because of our improved understanding, the amount of 
testing needed to prove stability has been reduced compared to that required 20 
to 40 years ago. 




(9-2) 



364 LIQUID PROPELLANT COMBUSTION AND ITS STABILITY 



PROBLEMS 



1 . For a particular liquid propellant thrust chamber the following data are given: 



Assume the gas composition and temperature to be uniform in the cylindrical chamber 
section. State any other assumptions that may be needed. Determine the approximate 
resonance frequencies in the first longitudinal mode, radial mode, and tangential mode. 

2. Explain how the three frequencies from Problem 1 will change with combustion tem- 
perature, chamber pressure, chamber length, chamber diameter, and throat diameter. 

3. Why does heat transfer increase during combustion instability? 

4 . Prepare a list of steps for undertaking a series of tests to validate the stability of a new 
pressure-fed liquid bipropellant rocket engine. State the assumptions. 

5. Estimate the resonant frequency of a set of each of nine cavities similar to Fig. 9-7. 
Here the chamber diameter D = 0.200 m, the slot width is 1.0 mm, and the width and 
height of the cavity are each 20.0 mm. The walls separating the individual cavities are 
10.0 mm thick. Assume L = 4.00 mm, AL = 2.00 mm, and a = 1050 m/sec. 



REFERENCES 

9-1. R. D. Sutton, W. S. Hines, and L. P. Combs, “Development and Application of a 
Comprehensive Analysis of Liquid Rocket Combustion,” AIAA Journal, Vol. 10, 
No. 2, February 1972, pp. 194-203. 

9-2. K. K. Kuo, Principles of Combustion, 2nd ed., John Wiley & Sons, Hoboken, NJ, 
2005. 

9-3. V. Yang and W. Anderson (Eds.), Liquid Rocket Engine Combustion Instability, 
Vol. 169 of Progress in Astronautics and Aeronautics , AIAA, 1995, in particular 
Chapter 1, F.E.C. Culick and V. Yang, “Overview of Combustion Instabilities in 
Liquid Propellant Rocket Engines.” 

9-4. W. Sirigano, A. Merzhanov, and L. De Luca, Advances in Combustion Science: 
In Honor of Ya. B. ZeVdovich, Progress in Astronautics and Aeronautics Series, 
V-173, American Institute of Aeronautics and Astronautics, 1997. 

9-5. D. T. Harrje (Ed.), “Liquid Propellant Rocket Combustion Instability,” NASA SP- 
194, U.S. Government Printing Office, No. 3300-0450, 1972. 

9-6. G. P. Sutton, History of Liquid Propellant Rocket Engines, AIAA, 2006, Chapter 
4.10 “Combustion and Vibration.” 



Chamber presure 

Chamber shape 

Internal chamber diameter 

Length of cylindrical section 

Nozzle convergent section angle 

Throat diameter and radius of wall curvature 

Injector face 

Average chamber gas temperature 
Average chamber gas molecular weight 
Specific heat ratio 



Cylindrical 
0.270 m 
0.500 m 
45° 

0.050 m 
Plat 
2800 K 
20 kg/kg-mol 
1.20 



68 MPa 



REFERENCES 365 



9-7. V. Yang, M. Habiballah, J. Hulka and M. Popp (Eds.), Liquid Rocket Thrust Cham- 
bers: Aspects of Modeling, Analysis and Design, Progress in Astronautics and 
Aeronautics (server), Vol 200, American Institute of Aeronautics and Astronautics, 
Reston, VA 2004. 

9-8. B. R. Lawyer, “Photographic Observations of Reactive Stream Impingement,” 
Journal of Spacecraft and Rockets, Vol. 17, No. 2, March- April 1980, 
pp. I34-I39. 

9-9. M. Tanaka and W. Daimon, “Explosion Phenomena from Contact of Hypergolic 
Liquids,” Journal of Propulsion and Power, Vol. 1, No. 4, 1984, pp. 314-316. 

9-10. R. I. Sujith, G. A. Waldherr, J. I. Jagoda, and B. T. Zinn, “Experimental Investiga- 
tion of the Evaporation of Droplets in Axial Acoustic Fields,” Journal of Propulsion 
and Power, AIAA, Vol. 16, No. 2, March-April 2000, pp. 278-285. 

9-11. P. Y. Liang, R. J. Jensen, and Y. M. Chang, “Numerical Analysis of the SSME 
Prebumer Injector Atomization and Combustion Process,” Journal of Propulsion 
and Power, Vol. 3, No. 6, November -December 1987, pp. 508-513. 

9-12. M. Habiballah, D. Lourme, and F. Pit, “PHEDRE — Numerical Model for Combus- 
tion Stability Studies Applied to the Ariane Viking Engine,” Journal of Propulsion 
and Power, Vol. 7, No. 3, May-June 1991, pp. 322-329. 

9-13. M. L. Dranovsky (author), V. Yang, F. E. C. Culick, and D. G. Talley (Eds.), Com- 
bustion Instability in Liquid Rocket Engines, Testing and Development; Practices 
in Russia, Progress in Astronautics and Aeronautics, Vol. 221, AIAA, Reston, VA, 
2007. 

9-14. M. S. Natanzon and F. E. C. Culick, Combustion Instability, Progress in Astro- 
nautics and Aeronautics, Vol. 222, AIAA, Reston, VA, 2008. 

9-15. B. W. Oppenheim and S. Rubin, “Advanced Pogo Analysis for Liquid Rockets,” 
Journal of Spacecraft and Rockets , Vol. 30, No. 3, May-June 1993, pp. 360-373. 

9-16. G. About, F. Bouvert, C. Bonna, N. David, and J. C. Lemoine, “A New Approach 
of POGO Phenomenon Three-Dimensional Studies on the Ariane 4 Launcher,” 
Acta Astronautica , Vol. 15, Nos. 6 and 7, 1987, pp. 321-330. 

9-17. K. W. Dotson, S. Rubin, and B. M. Sako, “Mission Specific Pogo Stability Analysis 
with Correlated Pump Parameters”, Journal of Propulsion and Power, Vol. 21, 
No. 4. July-August 2005, pp. 619-626. 

9-18. T. Shimura and K. Kamijo, “Dynamic Response of the LE-5 Rocket Engine Liquid 
Oxygen Pump,” Journal of Spacecraft and Rockets, Vol. 22, No. 7, March-April 
1985, pp. 195-200. 

9-19. E. Kirner, W. Oechslein, D. Thelemann, and D. Wolf, “Development Status of the 
Vulcain (HM 60) Thrust Chamber” AIAA Paper 90; 2255, July 1990. 

9-20. “Guidelines for Combustion Stability Specifications and Verification Procedures for 
Liquid Propellant Rocket Engines,” CPIA Publication 655, Chemical Propulsion 
Information Agency, John Hopkins University, January 1997. 

9-21. T. L. Acker and C. E. Mitchell, “Combustion Zone-Acoustic Cavity Interac- 
tions in Rocket Combustors,” Journal of Propulsion and Power, Vol. 10, No 2, 
March-April 1994, pp. 235-243. 



CHAPTER 10 



TURBOPUMPS AND THEIR GAS 
SUPPLIES 



10.1. INTRODUCTION 

This is the first revised edition of this book, which has a separate chapter on tur- 
bopumps (abbreviated here as TP) and their gas supplies. A TP is a high-pressure 
high-precision piece of rotating high-speed power machinery usually consisting 
of a turbine driving one or two centrifugal propellant pumps. Its purpose is to 
take propellants from the vehicle’s propellant tanks, raise their pressure, and 
deliver these propellants into the high-pressure piping system. The pressurized 
propellants are then fed to one or more thrust chambers, where they are burned 
and form hot gases. A TP is a unique key component of all larger liquid propel- 
lant rocket engines with a pumped feed system and its design, fabrication, and 
functions are unlike any of the other rocket engine components. This chapter dis- 
cusses the various common types of TPs, outlines one design approach, describes 
major TP components, and mentions some of their principal design issues. 

A well-designed TP delivers the intended propellant flows at the intended 
pump discharge pressures and mixture ratio, must have good enough reliabil- 
ity so that it will not malfunction or fail during the intended flight duration, 
and many will run at the highest practical energy efficiency of the pumps and 
turbines. Furthermore the TP should not stimulate or cause any significant vibra- 
tion (in the engine or vehicle) or should not be adversely affected by externally 
caused vibrations, will function well over all operating conditions (such as dif- 
ferent initial propellant temperatures, a range of ambient temperatures, start and 
stop transients, accelerations in the flight direction and other directions) at the 



366 



10.2. DESCRIPTIONS OF SEVERAL TURBOPUMPS 367 



minimum practical inert TP mass. It is important that the pumps will not cavitate 
because the cavitation bubbles can reduce the steady nominal propellant flow and 
can cause combustion instability. Cavitation is discussed later in Section 10.5. 
The bearings and seals need to be adequately cooled by small secondary flows 
of propellant within the TP assembly to prevent their overheating and malfunc- 
tioning. There should not be any unexpected leakage in the seals and secondary 
flow passages within a TP; this means leaks between bipropellants, between tur- 
bine stages (gas) or pump stages (liquid), between a propellant and hot gas, or 
between propellant or hot gas and the outside. 

The selection of a specific TP configuration, such as those seen later in this 
chapter, will depend on the propellant combination, the desired flows and pump 
discharge pressures, the engine cycle, the available suction pressure, the number 
of units to be built and delivered, and other factors. The available proven heat- 
resistant materials and their maximum working temperatures for turbine blades 
or the maximum load capacity of a bearing can influence the design selection. 
The best placement of a TP within a rocket engine is usually a compromise 
between several design considerations. For example, its inlet and outlet flanges 
should be placed to minimize the piping, access bearings and turbine blades 
for inspection on reusable engines, mounting which allows thermal expansion 
without high stresses, or the simplicity of the design (e.g., minimum number of TP 
components) — all these will influence the choice of the TP design configuration. 
References 10-1 to 10-5 give further information on TP selection and design. 



10.2. DESCRIPTIONS OF SEVERAL TURBOPUMPS 

Typical TPs can be seen in Figs. 10-1 to 10-3 and 6-10, and many key compo- 
nents can be identified by their callouts. Most of the part names in these figures 
will be discussed in subsequent paragraphs of this chapter. TPs are also seen as a 
component of an engine in Figs. 6-1, 6-11, 6-12, and 8-17. Figure 6-2 gives 
a diagram of various common feed system arrangements; many have one or more 
TPs. A number of different arrangements of the key TP components for several 
common TP assemblies are shown in Fig. 10-4. This includes different numbers 
of pumps and turbines, different component arrangements on TP shafts, with or 
without gear cases, or with and without booster pumps. All will be discussed 
later in this chapter. Figures 10-1 and 10-2 are experimental TPs. They were 
selected for this book because they clearly identify the key components, (see Ref. 
10-6). Figure 10-1 shows a single-stage propellant pump; it has a screw-type 
axial-flow inducer impeller ahead of the main impeller, is driven by a single- 
stage axial-flow turbine. The hot combustion gases, which drive this turbine, are 
burned in a separate gas generator (or a precombustion chamber or prebumer) 
at a mixture ratio that gives gases between 900 and 1200 K; this is sufficiently 
cool, so that the hot turbine hardware (blades, nozzles, manifolds, or disks) still 
have sufficient strength without needing forced cooling. The gases are expanded 
(accelerated) in an annular set of converging -diverging supersonic turbine inlet 



368 TURBOPUMPS AND THEIR GAS SUPPLIES 




Entry slots 
to turbine 
nozzles 



Turbine gas 
discharge flange 



Typical nozzle 



Turbine blades 
Turbine disk 
Shaft seal 



Angular ball 
bearing for 
axial loads 



Inducer 

impeller 



Pump diffuser 
vanes 



Hydrostatic 
bearing 

Internal 
passages for 
supplying propellant 
to bearings 

Pump volute with 
changing cross 
section 

Shrouded 
impeller 



bearing 



Nut holding 
inducer to 
shaft 



Pump 

suction 

flange 



Pump 

housing 



Gas inlet 
manifold and 
turbine housing 



Pump discharge 
flange 



FIGURE 10-1. Cut-away view of an experimental turbopump demonstrator with a 
single-stage liquid oxygen pump impeller, an inducer impeller, and a single-stage turbine 
(one row of blades) on the same shaft. (Courtesy of Pratt & Whitney Rocketdyne.) 



nozzles, which are usually cast into the cast turbine inlet housing. The gases then 
enter a set of rotating blades, which are mounted on a rotating wheel or turbine 
disk. The blades essentially remove the tangential energy of the gas flow. The 
exhaust gas velocity exiting from the blades is relatively low and its direction 
is essentially parallel to the shaft. The pump is driven by the turbine through a 
shaft which is supported by two experimental hydrostatic bearing. The propellant 
enters the pump through an inducer, a special impeller where the pressure of the 
propellant is raised only slightly (perhaps 5 to 10% of the total pressure rise). 
This is just enough pressure so that there will be no cavitation as the flow enters 
the main pump impeller. Most of the kinetic energy given to the flow by the 
pump impeller is converted into hydrostatic pressure in the diffusers (the diffuser 
vanes are not clearly visible, since they are inclined) and/or volutes of the pump. 
The two hydrostatic bearings support the shaft radially. All bearings and shaft 
seals create heat as they run. They are cooled and lubricated by small flows of 
propellant, which are supplied from the pump discharge through drilled passages. 



10.2. DESCRIPTIONS OF SEVERAL TURBOPUMPS 369 



Housing for filtered 
bearing supply 

y 



Pump 

inlet 



Cast pump housing with integral 
crossover passages (Inconel 718) 

Pump outlet 



One-piece titanium rotor with inducer, two 
impellers, turbine, and bearing surfaces 



Split hydrostatic bearing 
housings (Incoloy 909) 

Radial in-flow turbine 




Cast turbine housing with 
vaneless internal volute 



Turbine discharge 
flange 



Turbine gas 
inlet flange 



FIGURE 10-2. Exploded view of an advanced high-speed, two-stage liquid hydrogen 
fuel pump driven by a radial flow turbine. (Used with permission of Pratt & Whitney 
Rocketdyne, a part of United Technologies Corp.; adapted from Ref 10-7.) 



One bearing (near the pump) is very cold and the other is hot, since it is close 
to the hot turbine. The angular ball bearing accepts the axial net loads from the 
unbalanced hydrodynamic pressures around the shrouded impeller, the inducer, 
and also the turbine blades or the turbine disk. 

A novel, experimental, high-speed, compact, and light weight liquid hydrogen 
turbopump is shown in Fig. 10-2 and in Ref. 10-6 and 10-7. It was intended 
to be used with an upper-stage hydrogen -oxygen rocket engine with a thrust of 
about 50,000 Ibf (22.4 kN). The unique single-piece titanium rotor turns nomi- 
nally at 166,700 rpm, has two machined sets of pump vanes, a machined inducer 
impeller, a set of machined radial inflow turbine blades, and radial as well as 
axial hydrostatic bearing surfaces. A small filtered flow of hydrogen lubricates 
the hydrostatic bearing surfaces. The cast pump housing has internal crossover 
passages between stages. The unique radial inflow turbine (3.2 in. dia.) produces 
about 5900 hp at an efficiency of 78%. The hydrogen pump impellers are only 
3.0 in. diameter and produce a pump discharge pressure of about 4500 psi at a 
fuel flow of 16 Ibm/sec and an efficiency of 67%. A high pump inlet pressure of 
about 100 psi is needed to assure cavitation-free operation. The turbopump can 
operate at about 50% flow (at 36% discharge pressure and 58% of rated speed). 
The number of pieces to be assembled is greatly reduced, compared to a more 
conventional turbopump, thus enhancing its inherent reliability. 

The geared turbopump in Fig. 10-3 has high turbine and pump efficiencies, 
the rotary speed of the two-stage turbine is higher than the pump shaft speeds, 
and the turbine is smaller than a comparable single-shaft TP. The auxiliary power 



370 TURBOPUMPS AND THEIR GAS SUPPLIES 




turbine 



inlet 
manifold 



Oxygen 
main impeller 



Fuel 

pump 

impeller 



Inducer 

impeller 



Fuel 

outlet 



Auxiliary 

hydraulic 

pump 



2-stage spur 
reduction gears 



Volutes 



FIGURE 10-3. Typical geared turbopump assembly similar to the one used on the RS-27 
engine (Delta I and II launch vehicles) with liquid oxygen and RP-1 propellants. (Courtesy 
of Praft & Whitney Rocketdyne.) 



package (e.g., hydraulic pump) was used in early applications. The precision ball 
bearings and seals on the turbine shaft can be seen, but the pump bearings and 
seals are not visible in this figure. 

Table 10-1 gives selected data on relatively large TPs for two large LOX/LH 2 
rocket engines. It shows that the main LOX pumps have single-stage impellers, 
while the main fuel pumps are multistage with two or three impellers in series. 
The Space Shuttle Main Engine (SSME) has axial-flow inducer impellers on its 
main pumps and also two booster pumps, which together raise the pressure of the 
flows going to the inlets of the respective main centrifugal pumps. The booster 
pump data is also listed. The Japanese LE-7 engine feed system does not use 
booster pumps and it features pump inducers ahead of the main impellers. The 
turbine blade shapes for both TPs are a combination of impulse turbines with 
some reaction turbines contours. These terms will be explained later. 




TABLE 10-1. Turbopump Characteristics 



U § 
Ij 00 
2 

U *0 

0 X 

1 O' 

I J 

^ C3 



J 



(D 

00 C 
CO D 

®0 

(u O 

S -o 

'O • = 

1 

_1 



>> 

U 



U "O 



U 



2 I + 

C3 



(N ^ 



m 

lO r- (N IT) 
C 5 00 'O 
<N 



a. 

H 

a. 

a. 

X 



X 



o 



o. 

a. 

K 



0^: 



a. 

H 

o 

a. 

K 



0. O 

&. .2 
N-1 X 
< 



'O- o o 
o 



o ^ 1 C“ 

in ^ 



Os (N r- <N 



O O (N 



a. 

s 



s & 6 

Q eS' Z 



2 

1 >. 

c/2 CJ 

2 c 

a .2 

o 

1> cC 
00 ^ 
S “ 

.C P. 

a s 

Q (2 



00 Q fsj O 



rn o 
oo 



Oi — 



cn r-- (N 



m o o o 



Cu 

e 



O- 2 
0) £ 

02 — 



o^ p o o 



' 

o £ 

0:2 — 



TO- O O o 

o m - 

cn VO ^ 

(N in 



2 TD 

■?. ^ 

s o 



^ ON 
NO (N 
<N 



in (N 
in NO 



O p o 
NO f^r ON 
<N 

in ^ 



ON o o 
NO (N (N 

o — 



# ^ 

cr _ O 



o. 

£ 



Z 






c 

£ 

'o 



a* s 

c/2 Q. 
flj flj 
£ ^ 
2 15 

l-i u. 

O 3 

H H 



371 



‘'Data courtesy of Pratt & Whitney Rocketdyne, at flight power level of 104.5% of design thrust. 
*Data courtesy of Mitsubishi Heavy Industries, Ltd. 

‘LPOTP, low-pressure oxidizer turbopump; HPFTP, high-pressure fuel turbopump. 

Boost impeller stage for oxygen flow to prebumers. 



372 TURBOPUMPS AND THEIR GAS SUPPLIES 



10.3. SELECTION OF TURBOPUMP CONFIGURATION 

The selection of the features, performance, and configuration of a specific new 
TP arrangement can be complex. First of all it depends on the engine require- 
ments for the particular flight application (thrust, propellant combination, mixture 
ratio, chamber pressure, duration, low cost, schedule, etc.). These engine criteria 
have to be evaluated, weighed, and prioritized. Furthermore it depends on the 
engine cycle (see Section 6.6 and Fig. 6-9), propellant physical properties (such 
as vapor pressure, density, or viscosity), the nominal flows and pump discharge 
pressures (sum of chamber pressure, hydraulic losses in valves, cooling jacket, 
injector, and pipes), minimum practical inert TP mass, high reliability (no flight 
failures allowed), available suction pressure, maximum and minimum propel- 
lant and hardware initial temperatures, and the components arrangement of the 
assembly. It is also influenced by the size of the TP, the location of the TPs 
inlet and outlet flanges relative to the thrust chamber inlets and the suction pipes 
outlets, maximum allowable turbine gas inlet temperature (which depends on the 
turbine material), number of hot starts during the life time, critical speed of the 
rotating assembly, or the simplicity of the design (e.g., fewest number of parts). 
If reuse is an engine requirement, then additional factors have to be considered, 
such as ease of access to bearings, seals, or turbine blades for inspection of wear 
or cracks. With throttling of the thrust the TP has to operate efficiently over a 
range of speeds. There can be other TP selection criteria. Some of the preliminary 
selection and design criteria are explained in Refs. 10-1 to 10-3 and 10-5 and 
basic texts on pumps are in Refs. 10-8 and 10-9. 

The shaft speed is related to the TP diameter or size and thus to the TP mass . 
The tip speed of the pump impeller blades or the speed of the turbine blades at 
their mean diameter can be the same at different diameters and rotational speeds. 
Obviously, the highest practical shaft speed gives the smallest diameter, and thus 
usually the lowest inert TP mass. This is often in conflict with other criteria, such 
as cavitation avoidance, which is likely to be more difficult if the impeller speed 
is increased. It will be discussed later in this section. 

The arrangements of the various key TP components into an assembly can be 
an important design criteria. Figure 10-4 shows several common arrangements. 
The two most common are to have one turbine with two propellant pumps on the 
same shaft (Figs. 10-4a and 10-4b) and two smaller separate TPs, one where 
a turbine drives an oxidizer pump and one where another turbine drives a fuel 
pump (Fig. 10-4e). In Fig. 10-4a the fuel pump inlet has a shaft going through 
it and that will affect the suction pressure. In the TP in Fig. I0-4b the turbine 
requires more seals. Although not discussed here, the placement and selection of 
bearings and seals influences the TP selection. 

Centrifugal pumps are used in all known production TPs, and these are basi- 
cally constant-volume flow devices (Refs. 10-3, 10-4, 10-5, 10-8, and 10-9). 
If the two propellants have similar densities (say within about 40%), such as 
nitrogen tetroxide and UDMH or liquid oxygen and kerosene and the volume 
flow of oxidizer and of fuel are similar, then the same type of impeller (running 



1 0.3. SELECTION OE TURBOPUMP CONPIGURATION 373 



a 



L^- 

Two pumps on same shaft 
with outboard turbine. 
Shaft goes through fuei 
pump inlet. 



G 




Direct drive with turbine in 
middle. Shaft goes through 
turbine discharge manifold. 




Two main pumps and two booster 
pumps, each with its own gas turbine. 

C 




High-speed turbine and 
fuel pump geared to 
lower speed oxidizer 
d pump. 




Two turbines, each with one pump 
Gas flow shown in parallel, 
(alternate is gas flow in series, 
first through one and then the 
0 other turbine). 




With gear case, turbine can 
run faster. The two pumps 
I have different speeds. 




Two in-line shafts with flexible 
g coupling C. 



G G 




Two main pumps and 
two booster pumps, each 
with own turbine. 



FIGURE 10-4. Simplified diagrams of several different component arrangements of tur- 
bopumps (TP); F, fuel pump; O, oxidizer pump; T, turbine; G, hot gas; C, shaft coupling; 
and GC, gear case. (Used with permission from Ref. 10-1.) 



at the same speed) can be used on both of them on a single shaft. If the densities 
are quite different (e.g., liquid oxygen and liquid hydrogen), a single shaft would 
not be practical or efficient. Instead two separate TPs are used — rotating at high 
speed for the fuel pump and lower speed for the oxidizer pump — thus giving 
good pump efficiencies. This corresponds to Fig. 10-4e. 

It is at times difficult to package a relatively large TP into an engine package. 
The two-TP configuration is often easier to integrate into an engine assembly, 



374 TURBOPUMPS AND THEIR GAS SUPPLIES 



both pumps will be able to run at a relatively high efficiency, the TP will have 
fewer seals, and since the shafts are shorter, they are lighter for the same shaft 
stiffness. 

Figure 10-4g shows two shafts, in line with each other, with a mechanical cou- 
pling between them (Ref. 10-10). In this example the fuel pump and the turbine 
are on one shaft and the oxidizer pump on the other. The two shafts are connected 
by a flexible coupling, and this allows considerable shaft misalignment, and as 
a result some more generous fabrication and assembly tolerances in the TP. The 
first two-shaft inline TP assembly was in the German V-2 rocket engine (1938) 
with a six pin/sleeve type of flexible coupling. The Soviet Union developed a 
curvic coupling serrated sleeve as a coupling and it was smaller and lighter. Ref- 
erences 10-1 and 10-4 show a section view of these two TPs. The United States 
did not produce a TP with two inline coupled shafts. When compared to a larger 
single shaft, the smaller spans between bearings on each of the two shafts results 
in substantially smaller shaft diameters, which in turn can reduce the mass of the 
shafts, the diameter and mass of pumps and pump housings, and the inertia of 
the rotary assembly and can allow some increases of the shaft speed. 

Figures 10-4d and 10-4f show TPs with a gear case to transmit the power 
from the turbine to the pumps (Ref. 10-11). The purpose is to allow the turbine 
to run at high speed and the pumps to rotate at lower speeds, where the efficien- 
cies are relatively high, thus minimizing the amount of gas generator propellant 
needed to drive the TP. Gear cases for TPs were common in the United States 
beginning in the early 1950s and one is still in use. In Fig. 10-4e the turbine 
runs at a higher speed and the two pumps run at lower, but somewhat different 
speeds. In Fig. 10-4f the fuel is liquid hydrogen; the fuel pump and the turbine 
run at a higher speed on the same shaft and the oxygen pump is geared down to 
a lower speed. 

Figures 10-4c and 10-4h represent two arrangements for using booster TPs 
in addition to the two main TPs. See Ref. 10-12. The purpose is to reduce 
the vehicle’s tank pressure and inert mass further, thus improving the vehicle 
performance. A typical booster TP provides about 10% of the propellant pressure 
rise and the respective main pump about 90%. Booster pumps typically use axial- 
flow impellers, and a few can operate with considerable cavitation in the leading 
edges of the booster impellers. With a booster pump the pressure at the inlet to the 
main pump is raised enough so cavitation does not occur in these higher pressure 
impellers. The low rotating speed causes the booster pumps to have a relatively 
large diameter. In some liquid propellant rocket engines they are larger than the 
main pump. Since pressures in booster pumps are low, the walls are thin, and 
the booster pump inert mass can be low. Most of the turbines of booster TPs are 
driven by gas expansion, but some are powered by high-pressure liquid propellant 
taken from the discharge of the main pump. The total number worldwide of rocket 
engines with booster pumps is small, perhaps a dozen. Booster pumps provide 
low suction pressure and good cavitation resistance to the main pumps of the 
engines, but the engines are more complex and more costly. 

Bearings and seals will be overloaded or fail if the shaft deflects (even by a 
small amount like 0.001 in. in some TPs). See Refs. 10-13 and 10-14. In order to 



1 0 . 3 . SELECTION OE TURBOPUMP CONPIGURATION 375 



minimize deflection of the shaft, the shaft has to be stiff (less deflection for a given 
side load) and that means large in diameter and heavy. The rotating assembly 
(turbine, pumps, seals, bearings) has to be carefully balanced. An unbalance 
causes a side load (which can be substantial at high rotational speed) and a 
deflection of the shaft. Both accurate static and dynamic balances are usually 
required during assembly. The operating speed of the rotating assembly is never 
chosen to be the same as the resonant or critical shaft speed, where deflections 
can become very high and the bearings, seals, or even the shaft can fail. The 
synchronous or critical speed is usually well above the operating speed. A stiffer 
shaft (larger diameter, heavier, a shorter distance between bearings, and less 
overhang beyond the bearing) gives a higher critical speed and allows the pumps 
to run at higher speed, which allows a lighter TP assembly. There are some TP 
where the operating speed is actually above the critical speed. Here the shaft 
momentarily goes trough the critical speed during startup and shutdown. 

There are usually several secondary flow passages in a TP. A bearing, rotating 
shaft seal, or impeller wear ring are all usually cooled and/or lubricated with a 
very small flow of propellant, which is heated in the process and can then be 
fed back to the suction side of a pump or dumped overboard. It requires a set 
of small external tubes or internal passages in the TP assembly to route several 
small flows for achieving this cooling and lubrication of rubbing surfaces. There 
is a good number of seals and bearings that need a small propellant supply and 
the distribution system for this propellant can be complex; some routes have to 
be throttled to the desired small flow, and a mixing of unlike propellants has to 
be avoided. The analysis of these secondary flow passages can be tedious. If a 
bearing or seal is close to a turbine, they will get warm (needs more coolant) and 
if they are pumping a cryogenic propellant, the adjacent hardware will become 
very cold. The clearances of these bearings or seals have to be so designed that 
the intended clearance is correct at the operating conditions. 

Bubbles in the flow of propellant are indeed unwanted. They can be introduced 
into the propellant feed pipes by extensive cavitation in the pumps, (see Refs. 
10-4 and 10-9) by leaks in low-pressure suction piping, or by improper priming 
or filling of the propellant lines before or during start. Bubbles reduce the effec- 
tive density of the propellant, which in turn reduces the mass flow and the mixture 
ratio and usually reduces the thrust level somewhat below the nominal value. If 
the intended operating mixture ratio was fuel rich and bubbles were to reduce 
the fuel flow, this mixture ratio will approach a stochiometric value; this creates 
a very hot gas and would most likely cause a failure of the cooling jacket in less 
than a second. A major concern with bubbles is that they can readily trigger com- 
bustion instability (see Chapter 9), which often leads to thrust chamber failure. 
The key design objectives are to select a high enough shaft speed (which gives 
the smallest and lightest TP, but is still not prone to cavitation), a low enough 
pump suction pressure (which gives the lightest vehicle tank mass, but this pro- 
motes pump cavitation), and an impeller configuration, which will be strong and 
efficient and prevent dangerous cavitation under all operating conditions. 

A balance of axial fluid pressure forces within the TP rotating assembly and 
its housings is necessary to prevent excessive axial hydraulic pressure forces 



376 TURBOPUMPS AND THEIR GAS SUPPLIES 



(on surfaces normal to the shaft) from overloading certain parts of the assembly 
(Refs. 10-1 and 10-15). Unbalance would cause a small axial movement of the 
rotating assembly (typically just a few thousands of an inch) and often intense 
rubbing of stationary and rotating parts. These pressure forces are created by 
the fluids flowing through the pump paths (increasing pressures) and the turbine 
gases flowing through moving and stationary sets of turbine blades (decreasing 
pressures), and by the radial placement of wear rings or seals. Some turbopumps 
have balancing pistons and some use the fluid pressures on the back side of an 
impeller or turbine disk to make an effective balance piston. Pressure on the 
piston is controlled by a slight axial movement, which changes clearances at the 
edge of the piston. During the transient operations (start, stop, change thrust), 
there typically are brief periods of unbalance caused by changes in the pressure 
distributions of the internal flows, but they are of short duration. Nevertheless 
there can be some rubbing or axial contact of a rotating and a stationery surface 
normal to the shaft during the brief periods of unbalance. If the axial rubbing 
contact persists, there will be some damage to the TP. 



10.4. FLOW, SHAFT SPEEDS, POWER, AND PRESSURE BALANCES 

The design of a TP requires the careful balancing of the propellant flows, the 
shaft speeds, and the power between the key pumps and turbines and the pressure 
distribution of the propellant along its flow paths. For a TP with two pumps and a 
gear transmission, the shaft speeds can be written as in Eq. 10-1. The relationship 
between shaft speeds, torques, and power can readily be seen by reviewing a flow 
diagram of a TP such as Fig. 1-4. With a gear case the relation between shaft 
speeds can be written as 



Here N is the shaft speed, the subscripts t, f, and o stand for turbine, fuel pump, 
and oxidizer pump, respectively. The a„ and af are the gear ratios for the oxidizer 
and fuel pumps. When there are no gears, then Ug and aj become 1.0. For a TP 
similar to Fig. 10-4a and 10-4b: 



Here Lh is the torque to overcome friction in bearings, seals, or auxiliaries. For 
a TP as in Figs.l0-4a and 10-4b, the power of the turbine Pt has to equal the 
sum of the powers of the pumps plus some losses. This power balance can also 
be expressed as the product of the torque L and the speed N: 



Nf = agNo = ajNf 



( 10 - 1 ) 



Nt — Nf — Ng and L, — Lg -\- Lj -\- Lh 



( 10 - 2 ) 




(10-3) 



Pt = L,Nt = LfNf + LgNg + Pb 



(10-4) 



10.4. FLOW, SHAFT SPEEDS, POWER, AND PRESSURE BALANCES 377 



Here Pt is the power for overcoming the friction in all the bearings and seals, 
and in some TP conhgurations also the power loss in gears, and auxiliaries (such 
as an oil pump for lubricating gears). The power used in bearing, seals, and 
auxiliaries is usually small. 

For the two separated TP scheme in Fig. 10-4e each of the TPs has its own 
internal losses. It is obvious that the speeds N and the power P and even the 
torques L must balance in this conhguration: 

{N,)o = {Np)o and {N,)f = {Np)f (10-5) 

{Pt)o = i.Pp)o + {Pb)c and = {Pp)f + (10-6) 

The subscripts t and p stand for turbine and pump. The pressure balance 
equation for the fuel line (the oxidizer line is the same) at a point just past the 
fuel pump discharge flange can be written as 



(P/)d = {Pf)s + (A/J/-)pump (10-7) 

~ (Ap)i-nain fuel system T Pi 
— ( Ap)gas generator fuel system T Pgg 



Here the fuel pump discharge pressure (pf)d equals the fuel pump suction pres- 
sure {App)s plus the pressure rise across the pump (Ap)pump- This discharge 
pressure in turn has to equal the chamber pressure pi plus all the pressure drops 
in the high-pressure main fuel flow system downstream of the pump and upstream 
of the chamber. This usually includes the pressure losses in the cooling jacket, 
the injector, the piping, and the open fuel valve. This pump discharge pressure 
furthermore has to equal the gas generator combustion pressure pgg and all the 
pressure losses between the fuel pump discharge and the gas generator combus- 
tion chamber. A similar pressure balance is needed for the oxidizer systems. 

The fourth type of balance is different from the other three (flow, pressure, and 
power balance); it is aimed at preventing certain high-pressure axial hydraulic 
loads and/or axial gas pressure loads from causing damage inside the TP assem- 
bly See Refs. 10-1, 10-4, and 10-15. These axial internal loads (side forces 
on impellers or turbine disks) can be very high, particularly in large sized TPs. 
Certain ball bearings are capable of withstanding considerable axial loads, in 
addition to the usual radial loads. The axial movement of the rotating assembly 
is limited by design and is very small. This method of controlling axial forces 
with a ball bearing is used in many TPs, but usually the maximum axial load 
of ball bearings is limited; therefore it is used mostly with small-diameter TPs 
or low-pressure TPs. A balance of axial fluid pressure forces within a large TP 
rotating assembly and its large stationary housing assembly can be accomplished 
by balance pistons. Some TPs have balancing pistons and some use the fluid 
pressures on the back side of an impeller or turbine disk to make an effective 
balance piston. Otherwise it would cause undesirable rubbing of certain parts 
of the rotating TP assembly with parts of the stationary TP housing assembly. 



378 TURBOPUMPS AND THEIR GAS SUPPLIES 



Unbalanced large high-pressure loads cause a very small axial movement of the 
rotating assembly (typically just 0.001 to 0.003 inch) and the balancing mech- 
anism can be usually designed to prevent further axial movement and intense 
rubbing of stationary and rotating parts. Pressure on the balance piston is con- 
trolled by a slight axial movement, which changes the circular clearances at the 
edge of the piston, and the rotating assembly is kept from moving more that 
a couple of thousands of an inch. During the transient operations (start, stop, 
change thrust) there typically are brief periods of unbalance caused by changes 
in the pressure distributions on the internal surfaces normal to the shaft, but they 
are of short duration. Nevertheless there can be some rubbing or axial contact of 
a rotating and a stationery surface normal to the shaft during the brief periods 
of unbalance. If the axial rubbing contact persists, there will usually be some 
damage to the TP. 

A phenomena characteristic of rotating machinery is the relative displacement 
and very small movement of the center of the rotating shaft within the con- 
finement of small clearances of a hxed sleeve bearing; this is known as whirl. 
Reference 10-16 describes a condition where the relative small movement of the 
shaft’s center is synchronous or the same as the rotation movement of the shaft 
itself (revolutions per unit time). 

The above equations relate to the steady-state operating condition. The tran- 
sient and dynamic conditions are important and many have been analyzed using 
iterative procedures and digital computers. Many have been tested, as shown 
in Ref. 10-17. They are the start, stop, and thrust change. For example, the 
transient starting condition includes the tank pressurization, the Tiling of pipes, 
pumps, or manifolds with liquid propellants, the filling of turbines and their man- 
ifolds with heated gas, ignition of gas generators, or thrust buildup. However, no 
detailed discussion of these transient conditions will be given in this book. These 
dynamic conditions can be complex, are related to the combustion behavior, and 
are sometimes difficult to analyze. Each major rocket propulsion organization has 
developed some method for analyzing these transients and they are often specific 
to specific engines or specific organizations. A similar yet simpler and different 
transient analysis of flows and pressure drops is usually performed for engines 
with a pressurized gas feed system. 



10.5. PUMPS 

Classification and Description 

The centrifugal pump is generally considered the most suitable for pumping 
propellant in medium-sized and large rocket units. For the large flows and high 
pressures involved, they are efficient as well as economical in terms of mass and 
space requirement. 

Figure 10-5 is a simplified schematic drawing of a centrifugal pump. Fluid 
entering the impeller, which is essentially a wheel with spiral curved vanes rotat- 
ing within a casing, is accelerated within the impeller channels and leaves the 



10.5. PUMPS 379 




Inlet flange 




FIGURE 10-5. Simplified schematic half cross section of a typical centrifugal pump. 



impeller periphery with a high velocity to enter the volute, or collector, and 
thereafter the dijfuser, where conversion from kinetic energy (velocity) to poten- 
tial energy (pressure) takes place. In some pumps the curved diffuser vanes are 
upstream of the collector. The three-dimensional hydraulic design of impeller 
vanes, diffuser vanes, and volute passages can be accomplished by computer 
programs to give high efficiency and adequate strength. Internal leakage, or 
circulation between the high-pressure (discharge) side and the low-pressure (suc- 
tion) side of an impeller, is held to a minimum by maintaining close clearances 
between the rotating and stationary parts at the seals or wear ring surfaces. 
External leakage along the shaft is minimized or prevented by the use of a shaft 
seal. Single-stage pumps (one impeller only) are stress limited in the pressure 
rise they can impart to the liquid, and multiple-stage pumps are therefore needed 
for high pump head,* such as with liquid hydrogen. References 10-2, 10-5, 
10-8, 10-9, and 10-17 give information on different pumps. There is a free 
passage of flow through the pump at all times, and no positive means for shutoff 
are provided. The pump characteristics, that is, the pressure rise, flow, and effi- 
ciency, are functions of the pump speed, the impeller, the vane shape, and the 
casing configuration. Figure 10-6 shows a typical set of curves for centrifugal 
pumps. The negative slope on the head versus flow curve indicates a stable pump 
behavior. Reference 10-7 describes the development of a smaller turbopump and 
the testing of a spiral high-speed first-stage impeller, called an inducer. 

A shrouded impeller has a shroud or cover (in the shape of a surface of 
revolution) on top of the vanes as shown in Figs. 10-1, 10-3, and 10-5. This 
type usually has higher stresses and lower leakage around the impeller. In an 



*Pump head means the difference between pump discharge and pump suction head. Its units are 
meters or feet. The head is the height of a column of liquid with equivalent pressure at its bottom. 
The conversion from pounds per square inch into feet of head is: (X) psi = 144 (,!(')/density (Ib/fO). 
To convert Pascals (N/m^) of pressure into column height (m), divide by the density (kg/m^) and 
go (9.806m/sec^). 






jaModasjoq d>|ejg 

I § I 8 

I I I I I I I I 

)33j ‘pean uoipns SAOjSod )3U psjjnbay 



—I o 




E 

k 



b 



1 I I I I I I I I I I 

88SSSS5gg2° 

" tu 30 j 9 d duinj 



FIGURE 10-6. Water test performance curves of the centrifugal pumps of the German V-2 rocket engine. The propellants are 75% 
alcohol and liquid oxygen. 



10.5. PUMPS 381 



unshrouded impeller or turbine the vanes are not covered as seen in the turbine 
vanes in Fig. 10-2. 

Pump Parameters 

This section outlines some of the important parameters and features that have 
to be considered in the design of rocket propellant centrifugal pumps under 
steady-flow conditions. 

The required pump flow is established by the rocket engine design for a given 
thrust, effective exhaust velocity, propellant densities, and mixture ratio. In addi- 
tion to the flow required by the thrust chamber, the propellant consumption of 
the gas generator, and in some designs also a bypass around the turbine and aux- 
iliaries have to be considered in determining the pump flows. The required pump 
discharge pressure is determined from the chamber pressure and the hydraulic 
losses in valves, lines, cooling jacket, and injectors (see Eq. 10-7). To obtain 
the rated flow at the rated pressure, an additional adjustable pressure drop for a 
control valve or orifice is usually included which permits a calibration adjustment 
or change in the required feed pressure. A regulation of the pump speed can also 
change the required adjustable pressure drop. As described in Section 11.5, this 
adjustment of head and flow is necessary to allow for hydraulic and performance 
tolerances on pumps, valves, injectors, propellant density, and so on. 

It is possible to predict the pump performance at various speeds if the perfor- 
mance is known at any given speed. Because the fluid velocity in a given pump 
is proportional to the pump speed N , the flow quantity or discharge Q is also 
proportional to the speed and the head H is proportional to the square of the 
speed. This gives the following relations: 

Q(flow) ~ N (rpm or rad/sec) 

77 (pump head) - (10-8) 

P(pump power) ~ 

From these relations it is possible to derive a parameter called the specific 
speed Ns- It is a dimensionless number derived from a dimensional analysis of 
pump parameters as shown in Ref. 10-8. 

Ns = Ns/&/(go A77,)3/4 (10-9) 

Any set of consistent units will satisfy the equation: for example, N in radians 
per second, Q in m^/sec, go as 9.8m/sec^, and 77 in meters. The subscript e refers 
to the maximum efficiency condition. In U.S. pump practice it has become the 
custom to delete go, express N in rpm, and Q in gallons per minute or ft^/sec. 
Much of the existing U.S. pump data is in these units. This leads to a modified 
form of Eq. 10-2, where Ns is not dimensionless, namely 

A, = 21.2A7a/(A77,)3/4 



( 10 - 10 ) 



382 TURBOPUMPS AND THEIR GAS SUPPLIES 



TABLE 10-2. Pump Types 

Impeller type 

Radial Francis Mixed Flow Near Axial Axial 



Basic shape 
(half section) 




Specific speed Ns 
U.S. nomenclature 
SI consistent units 

Efficiency % 



500-1000 1000-2000 2000-3000 
0.2-0.3 0.4 0.6-0.8 

50-80 60-90 70-92 



3000-6000 Above 8000 
1. 0-2.0 Above 2.5 

76-88 75-82 



Adapted from Ref. 10-8. 



The factor 21.2 applies when N is in rpm, Q is in ft^/sec, and H is in feet. For 
each range of specific speed, a certain shape and impeller geometry has proved 
most efficient, as shown in Table 10-2. Because of the low density, hydrogen 
can be pumped effectively by axial-flow devices. 

The impeller tip speed in centrifugal pumps is limited by design and material 
strength considerations to about 200 to 450 m/sec or roughly 655 to 1475 ft/sec. 
With titanium (lower density than steel) and machined unshrouded impellers a 
tip speed of over 2150 ft/sec is now possible and used on the pump shown in 
Fig. 10-2. For cast impellers this limiting value is lower than for machined 
impellers. This maximum impeller tip speed determines the maximum head that 
can be obtained from a single stage. The impeller vane tip speed u is the product 
of the shaft speed, expressed in radians per second, and the impeller radius and 
is related to the pump head A.H by 

u = ir^lgQ AH (10-11) 

where the velocity factor tj/ has values between 0.90 and 1.10 for different 
designs. For many pumps, = 1.0. 

The volume flow Q defines the impeller inlet and outlet areas according to the 
equation of continuity. The diameters obtained from this equation should be in 
the proportion indicated by the diagrams for a given specific speed in Table 10-2. 
The continuity equation for an incompressible liquid is 

Q=Aivi=A 2 V 2 (10-12) 

where the subscripts refer to the impeller inlet and outlet sections, all areas A 
being measured normal to their respective flow velocity v. The inlet velocity m 
ranges usually between 2 and 6 m/sec or 6.5 to 20 ft/sec and the outlet velocity 



10.5. PUMPS 383 



V 2 between 3 and 15 m/sec or 10 to 47 ft/sec. For a compressible liquid, such as 
liquid hydrogen, the density p will change with pressure. The continuity equation 
then is in terms of mass flow in 



m = Aivipi = A 2 V 2 P 2 (10-13) 

The head developed by the pump will then also depend on the change in density. 

The pump performance is limited by cavitation, a phenomenon that occurs 
when the static pressure at any point in a fluid flow passage becomes less than 
the fluid’s vapor pressure. Cavitation is discussed in Refs. 10-1 to 10-5, 10-8, 
10-9, and 10-17. The formation of vapor bubbles in the low-pressure regions 
causes cavitation. These bubbles collapse when they reach a region of higher 
pressure, that is, when the static pressure in the fluid is above the vapor pressure. 
In centrifugal pumps cavitation is most likely to occur behind the leading edge of 
the pump impeller vane at the inlet because this is the point at which the lowest 
absolute pressure is encountered. The excessive formation of vapor causes the 
pump discharge mass flow to diminish and fluctuate and can reduce the thrust 
and make the combustion erratic and dangerous. 

When the bubbles travel along the pump impeller surface from the low- 
pressure region (where they are formed) to the downstream higher-pressure 
region, the bubbles collapse. The sudden collapses create local high-pressure 
pulses that have caused excessive stresses in the metal at the impeller surface. 
In most rocket applications this cavitation erosion is not as serious as in water 
or chemical pumps because the cumulative duration is relatively short and the 
erosion of metal on the impeller is not usually extensive. However, it has been 
a concern with some test facility transfer pumps. 

The required suction head {Hs)r is the limit value of the head at the pump inlet 
(above the local vapor pressure); below this value cavitation in the impeller may 
occur. It is a function of the pump and impeller design and its value increases 
with flow as can be seen in Fig. 10-6. To avoid cavitation the suction head 
above the vapor pressure required by the pump {Hs)r must always be less than 
the available or net positive suction head furnished by the line up to the pump 
(Hs)a, that is, {Hs)r < {Hs)a- The required suction head above vapor pressure 
can be determined from the suction speciflc speed S : 

S =2l.2N^e/(H,)T (10-14) 

The suction speciflc speed S depends on the quality of design and the speciflc 
speed Ns, as shown in Table 10-2. The suction speciflc speed S has a value 
between 5000 and 60,000 when using ft-lbf units. For pumps with poor suc- 
tion characteristics it has values near 5000, for the best pump designs without 
cavitation it has values near 10,000 and 25,000, and for pumps with limited 
and controllable local cavitation it has values above 40,000. In Eq. 10-14 the 
required suction head {Hs)r is usually defined as the critical suction head at 
which the developed pump discharge head has been diminished arbitrarily by 



384 TURBOPUMPS AND THEIR GAS SUPPLIES 



2% in a pump test with increasing throttling in the suction side. Turbopump 
development has, over the last several decades, led to impeller designs which 
can operate successfully with considerably more cavitation than the arbitrary and 
commonly accepted 2% head loss limit. Inducers are now designed to run stably 
with extensive vapor bubbles near the leading edge of their vanes, but these bub- 
bles collapse at the trailing end of these vanes. Inducers now can have S values 
above 80,000. A discussion of one method for the design of impeller blades can 
be found in Ref. 10-8. 

The head that is available at the pump suction flange is called the net positive 
suction head or available suction head above vapor pressure {Hs)a- It is an 
absolute head value determined from the tank pressure (the absolute gas pressure 
in the tank above the liquid level), the elevation of the propellant level above 
the pump inlet, diminished by the friction losses in the line between tank and 
pump, and the vapor pressure of the fluid. When the flying vehicle is undergoing 
accelerations, the head due to elevation must be corrected accordingly. These 
various heads are defined in Fig. 10-7. The net positive suction head (Hs)a is 
often abbreviated as NPSH and is the maximum head available for suppressing 
cavitation at the inlet to the pumps; 

{bls)A — ^tank “f ^elevation ^friction ^vapor (10—15) 



Tank pressure 
gauge (SJ 



Fluid 



level 




Atmospheric 
pressure 

^ Absolute tank_ 
pressure, Huuj, 



Gauge tank 
gas pressure 






Typical |i/=^ 
valve “>= 



Pump inlet 




Jl 



Friction loss Hr 



Absolute static 
head at pump 
inlet with 
zero flow 



Elevation 

head 



^ Atmospheric 
^ pres^sure 



Friction loss 



T 



Vapor pressure 
of fluid, 



Available gauge 
pressure at 
pump inlet 
with flow 



Absolute available 
dynamic and 
static head at 
pump inlet 
with flow 



Net positive suction 
head (NPSH) or 
maximum head 
available for 
surpressing cavitation 
In the pump 
(Hs)x 



flange 



Pressure reference line 



Pump 



FIGURE 10-7. Definition of pump suction head. 



10.5. PUMPS 385 



To avoid pump cavitation, {Hs)a has to be higher than (Hs)r. If additional head 
is required by the pump, the propellant may have to be pressurized by external 
means, such as by the addition of another pump in series (called a booster pump) 
or by gas pressurization of the propellant tanks. This latter method requires thicker 
tank walls and, therefore, heavier tanks, and a bigger gas-pressurizing system. 
For example, the oxygen tank of the German V-2 was pressurized to 2.3 atm, 
partly to avoid pump cavitation. For a given value of {Hs)a, propellants with 
high vapor pressure require correspondingly higher tank pressures and heavier 
inert tank masses. For a given available suction head {Hs)a, a pump with a low 
required suction pressure usually permits designs with high shaft speeds, small 
diameter, and low pump inert mass. A small value of {Hs)r is desirable because it 
may permit a reduction of the requirements for tank pressurization and, therefore, 
a lower inert tank mass. The value of (Hs)r will be small if the impeller and 
fluid passages are well designed and if the shaft speed N is low. A very low 
shaft speed, however, requires a large-diameter pump, which will be excessively 
heavy. The trend in selecting centrifugal pumps for rocket application has been 
to select the highest shaft speed that gives a pump with a low value of (Hs)r, 
does not require excessive tank pressurization or other design complications, and 
thereby permits relatively lightweight pump design. This places a premium on 
pumps with good suction characteristics. 

There have been some low-thrust, low-flow, experimental engines that have 
used positive displacement pumps, such as diaphragm pumps, piston pumps, or 
rotary displacement pumps (gear and vane pumps). For low values of specific 
speed Ns these pumps have much better efficiencies, but their discharge pressures 
fluctuate with each stroke and they are noisy. 

One method to provide a lightweight turbopump with low vehicle tank pressure 
is to use an inducer, which is a special pump impeller usually on the same shaft 
and rotating at the same speed as the main impeller. It has a low head rise 
and therefore a relatively high specific speed. Inducer impellers are immediately 
upstream of the main impeller. They are basically axial-flow pumps with a spiral 
impeller, and many will operate under slightly cavitating conditions. The inducer 
stage’s head rise (typically, 2 to 10% of the total pump head) has to be just large 
enough to suppress cavitation in the main pump impeller; this allows a smaller, 
lighter, higher-speed main pump. Figures 10-2 and 10-8 and Reference 10-18 
show inducers. Reference 10-19 describes the testing of a pump with an inducer. 
Most TPs today have inducers. 



Influence of Propellants 

For the same power and mass flow, the pump head is inversely proportional to 
the propellant density. Since pumps are basically constant-volume flow machines, 
the propellant with the highest density requires less head, less power, and thus 
allows a smaller pump assembly. 

Because many of the propellants are dangerous to handle, special provi- 
sions have to be made to prevent any leakage through the shaft seals. With 



386 



TURBOPUMPS AND THEIR GAS SUPPLIES 




FIGURE 10-8. Fuel pump inducer impeller of the Space Shuttle main engine low- 
pressure fuel turbopump. It has a diameter about 10 in., a nominal hydrogen flow of 
148.6 Ibm/sec, a suction pressure of 30 psi, a discharge pressure of 280 psi at 15,765 
rpm, an efficiency of 77%, and a suction specific speed of 39,000 when tested with water. 
(Courtesy of Pratt & Whitney Rocketdyne.) 



spontaneously ignitable propellants the leakages can lead to fires in the pump 
compartment and may cause explosions. Multiple seals are often used with a 
drainage provision that safely removes or disposes of any propellants that flow 
past the first seal. Inert-gas purges of seals have also been used to remove haz- 
ardous propellant vapors. The sealing of corrosive propellants puts very severe 
requirements on the sealing materials and design. With cryogenic propellants 
the pump bearings are usually lubricated by the propellant, since lubricating oil 
would freeze at the low pump hardware temperature. 

Centrifugal pumps should operate at the highest possible pump efficiency. This 
efficiency increases with the volume flow rate and reaches a maximum value of 
about 90% for very large flows (above 0.05 m^/sec) and specific speeds above 
about 2500 (see Refs. 10-1 and 10-8). Most propellant pump efficiencies are 
between 30 and 70%. The pump efficiency is reduced by surface roughness 
of casing and impellers, the power consumed by seals, bearings, and stuffing 
boxes, and by excessive wear ring leakage and poor hydraulic design. The pump 
efficiency rjp is defined as the fluid power divided by the pump shaft power Pp -. 

r]p=pQAH/Pp (10-16) 

A correction factor of 550 ft-lbf/hp has to be added if Pp is given in horsepower, 
the head H in feet, and the volume flow Q in ft^ /sec. When using propellants, 
the pump power has to be multiplied by the density ratio if the required power 
for water tests is to be determined. 



10 . 6 . TURBINES 387 



Example 10-1. Determine the shaft speed and the overall impeller dimensions for a 
liquid oxygen pump which delivers 500 Ibm/sec of propellant at a discharge pressure of 
1000 psia and a suction pressure of 14.7 psia. The oxygen tank is pressurized to 35 psia. 
Neglect the friction in the suction pipe and the suction head changes due to acceleration 
and propellant consumption. The initial tank level is 15 ft above the pump suction inlet. 

SOLUTION. The density of liquid oxygen is 71.2 Ibm/ft^ at its boiling point (from 
Chapter 7). The volume flow will be 500/71.2 = 7.022 ft^/sec. The vapor pressure of the 
oxygen is 1 atm = 14.7 psi = 29.8 ft. The suction head is 35 x 144/71.2 = 70.8 ft. From 
Eq. 10-12 the available suction head is 70.8 + 29.8 = 100.6 ft. The available suction head 
above vapor pressure is {Hs)a = 70.8 + 15.0 — 0 — 29.8 = 56.0 ft. The discharge head 
is 1000 X 144/71.2 = 2022 ft. The head delivered by the pump is then 2022 — 100.6 = 
1921 ft. 

The required suction head will be taken as 80% of the available suction head in order 
to provide a margin of safety for cavitation (Hs)r = 0.80 x 100.6 = 80.48 ft. Assume a 
suction specific speed of 15,000, a reasonable value if no test data are available. From 
Eq. 10-14 solve for the shaft speed N\ 

S = = 21.2AV7.022/80.48°''^ = 15,000 

Solve for N = 7174 rpm or 751.3 rad/sec. 

The specific speed, from Eq. 10-9, is 

A, = 2\.2Ny[QIH^I‘' = 21.2 x 7174V7.022/192l“-’^ = 1388 

According to Table 10-2, the impeller shape for this value of A, will be a Francis type. 
The impeller discharge diameter D 2 can be evaluated from the tip speed by Eq. 10-11: 

u = fy/2go AH = I.OV 2 X 32.2 x 1921 = 352 ft/sec 

D 2 = 353 X 2/664.7 = 1.062 ft = 12.75 in. 

The impeller inlet diameter Dj can be found from Eq. 10-5 by assuming a typical inlet 
velocity of 15 ft/sec and a shaft cross section 5.10 in.^ (2.549 in. diameter). 

A = Q/vi = 7.022/15 = 0.468 ft^ = 67.41 in.^ 

A = ijrDf = 67.41 + 5.10 = 72.51 in.^ 

Di = 9.61 in. (internal flow passage diameter) 

This is enough data to draw a preliminary sketch of the impeller. 



10.6. TURBINES 

The turbine must provide adequate shaft power for driving the propellant pumps 
(and sometimes also auxiliaries) at the desired shaft speed and torque. The turbine 
derives its energy from the expansion of a gaseous working fluid through fixed 



388 TURBOPUMPS AND THEIR GAS SUPPLIES 



nozzles and rotating blades. The blades are mounted on disks to the shaft. The gas 
is expanded to a high, nearly tangential, velocity and through inclined nozzles and 
then flows through specially shaped blades, where the gas energy is converted 
into tangential forces on each blade. These forces cause the turbine wheel to 
rotate (see Refs. 10-1 to 10-5, 10-20 and 10-21). 

Classification and Description 

The majority of turbines have blades at the periphery of a turbine disk and the 
gas flow is axial, similarly in concept to the axial-flow pattern shown for pumps 
in Table 10-2 and the single-stage turbine of Fig. 10-1. However, there are a 
few turbines with radial flow (particularly at high shaft speeds), such as the one 
shown in Fig. 10-2. Ideally there are two types of axial-flow turbines of interest 
to rocket pump drives: impulse turbines and reaction turbines, as sketched in 
Fig. 10-9. In an impulse turbine the enthalpy of the working fluid is converted 
into kinetic energy within the first set of stationary turbine nozzles and not in 
the rotating blade elements. High-velocity gases are delivered (at a small angle 
to a tangential direction) to the rotating blades, and blade rotation takes place as 
a result of the impulse imparted by the momentum of the fluid stream of high 
kinetic energy to the rotating blades which are mounted on the turbine disk. The 
velocity-staged impulse turbine has a stationary set of blades which changes the 
flow direction after the gas leaves the first set of rototating blades and directs 
the gas to enter a second set of rotating blades in which the working fluid gives 
up further energy to the turbine wheel. In a pressure-staged impulse turbine, the 
expansion of the gas takes place in all the stationary rows of blades. In a reaction 
turbine the expansion of the gas is roughly evenly split between the rotating and 
stationary blade elements. The high-pressure drop available for the expansion of 
the turbine working fluid in a gas generator cycles favors simple, lightweight 
one- or two-stage impulse turbines for high thrust engines. Many rocket turbines 
are neither pure impulse nor reaction turbines, but often are fairly close to an 
impulse turbine with a small reaction in the rotating vanes. 

With some gas generator engine cycles the turbine exhaust gases pass through 
a De Laval Supersonic nozzle at the exit of the exhaust pipe (see Fig. 1-4). The 
high turbine outlet pressure gives critical flow conditions at the venturi throat 
(particularly at high altitudes) and thereby assures a constant turbine outlet pres- 
sure and a constant turbine power which will not vary with altitude. Furthermore, 
it provides a small additional thrust to the engine. 

Turbine Performance and Design Considerations 

The power supplied by the turbine is given by a combined version of Eqs. 3-1 
and 3-7: 

P, = t]Tmt/Sh (10-17) 

P, = rnmrCpTdl - (10-18) 



10 . 6 . TURBINES 389 




impulse turbine 




FIGURE 10-9. Top view diagram, pressure and velocity profiles, and efficiency curves 
for impulse and reaction type turbines. The velocity ratio is the pitch line velocity of the 
rotating blades u divided by the theoretical gas spouting velocity co derived from the 
enthalpy drop. (Adapted with permission from Refs. 10-3 and 10-19.) 



The power delivered by the turbine Pt is proportional to the turbine efficiency 
rit, the mass flow through the turbine tii,, and the available enthalpy drop per 
unit of flow A/t. The units in this equation have to be consistent (1 Btu = 
778 ft-lbf = 1055 J). This enthalpy is a function of the specific heat Cp, the 
nozzle inlet temperature Ti, the pressure ratio across the turbine, and the ratio of 
the specific heats k of the turbine gases. For gas generator cycles the pressure 
drop between the turbine inlet and outlet is relatively high, but the turbine flow 
is small (typically 2 to 5% of full propellant flow). For staged combustion cycles 
this pressure drop is very much lower, but the turbine flow is much larger. 

For very large liquid propellant engines with high chamber pressure the turbine 
power can reach over 250,000 hp, and for small engines this could be perhaps 
around 35 kW or 50 hp. 



390 TURBOPUMPS AND THEIR GAS SUPPLIES 



According to Eq. 6-12, the power delivered by the turbine Pt has to be 
equal to the power required by the propellant pumps, the auxiliaries mounted on 
the turbopump (such as hydraulic pumps, electric generators, tachometers, etc.), 
and power losses in hydraulic friction, bearings, gears, seals, and wear rings. 
Usually these losses are small and can often be neglected. The effect of the 
turbine gas flow on the specific impulse of the rocket engine system is discussed 
in Section 6.6. For gas generator engine cycles, the rocket designer is interested 
in obtaining a high turbine efficiency and a high turbine inlet temperature T\ in 
order to reduce the flow of turbine working fluid, and for gas generator cycles 
also to raise the overall effective specific impulse, and, therefore, reduce the 
propellant mass required for driving the turbine. Three-dimensional computer 
analyses of the gas flow behavior and turbine blade geometry have resulted in 
efficient blade designs. 

Better turbine blade materials (such as single crystals which have been uni- 
directionally solidified) and specialty alloys can allow turbine inlet temperatures 
between 1400 K (or about 2050°F) and perhaps 1600 K (or 2420°F); these higher 
temperatures or higher gas enthalpies reduce the required turbine flow. Reliabil- 
ity, gas temperature variations or nonuniformity, and cost considerations have 
kept actual turbine inlet temperatures at conservative values, such as 1150 to 
1250°F or about 900 to 950 K, using special steel alloys as the blade and disk 
materials. The efficiency of turbines for rocket turbopumps is shown in Fig. 10-9. 
Maximum blade speeds with good design and strong high-temperature materials 
are typically 400 to 700 m/sec or about 1300 to 2300 ft/sec. Fligher blade speeds 
generally allow an improvement in efficiency. For the efficiency to be high the 
turbine blade and nozzle profiles have to have smooth surfaces. Small clearances 
at the turbine blade tips are also needed to minimize leakage around the blades. 

Different organizations define turbine efficiency in different ways. One way 
to define the turbine efficiency ip is the turbine power output LN divided by the 
turbine input from an ideal uniform heated gas flowing through an ideal turbine. 
The gas flow can be considered the ideal isentropic enthalpy drop. The actual 
pressure drop and the actual Ah or usually higher than the ideal: 



T], =L,N,/rn Ah (10-19) 

Flere Ah is the enthalpy drop with isentropic gas expansion across the turbine 
nozzles and the turbine buckets or blades (moving and stationary) per unit gas 
mass flow. It is typically based on a uniform gas flow across a flow section, 
uniform gas properties across a flow section, perfect gases, no leakage or alter- 
natively a nominal leakage around the blades, and perfect blade contours. Even a 
small clearance at the tip of turbine blades causes substantial losses, particularly 
at small turbine diameter and small blade heights. Some organizations include a 
few unavoidable basic losses in defining the ideal power. 

The advantage of increased turbine efficiency (less gas generator propellant 
requirement) can be realized only if the turbopump design has efficient turbine 
blade contours and allows high blade speeds. This speed can be achieved in 



10.7. APPROACH TO TURBOPUMP PRELIMINARY DESIGN 391 



rockets of medium and low thrust by gearing the turbine to the pump shaft or 
by using pumps that permit high shaft speeds; in rockets of very high thrust the 
pumps have diameters and shaft speeds close to those of the turbines and can 
be mounted on the same shaft as the turbine. The power input to the turbine can 
be regulated by controlling the gas flow and gas temperatures to the turbine inlet 
through valves or orifices. 

Usually, the preliminary analysis for the pump is done first. Avoiding exces- 
sive cavitation sets a key pump parameter, namely the maximum pump shaft 
speed. This is the highest possible shaft speed, which in turn allows the lightest 
turbopump mass, without excessive cavitation in the pump. If excessive cavita- 
tion occurs at the leading edge of the main impeller, then the flow will become 
unsteady and variable, leading to lower thrust and possible combustion instabil- 
ity. The amount of pressure in the vehicle (gas pressure in propellant tank plus 
the static elevation pressure) that can be made available to the engine (at the 
pump inlet) for suppressing cavitation has to be larger than the impeller vanes’ 
own pressure limit to cavitate. This allows us then to determine the shaft speed, 
which in turn can establish the approximate pump efficiencies, impeller tip speed 
(usually also limited by the material strength of the impeller), number of pump 
stages, key dimensions of the impeller. 



10.7. APPROACH TO TURBOPUMP PRELIMINARY DESIGN 

With all major rocket engine components the principal criteria (high performance 
or efficiency, minimum mass, high reliability, and low cost) have to be weighted 
and prioritized for each vehicle mission. For example, high efficiency and low 
mass usually mean low design margins, and thus lower reliability. A higher 
shaft speed will allow a lower mass turbopump, but it cavitates more readily 
and requires a higher tank pressure and heavier vehicle tanks (which usually 
outweighs the mass savings in the turbopump). 

The engine requirements give the initial basic design goals for the preliminary 
design of the turbopump, namely propellant flow, the pump outlet or discharge 
pressure (which has to be equal to the chamber pressure plus the pressure drops in 
the piping, valves, cooling jacket, and injector), the best engine cycle (as shown 
in Fig. 6-9), the start delay, and the need for restart or throttling, if any. Also, the 
propellant properties (density, vapor pressure, viscosity, or boiling point) must 
be known. Some of the design criteria are explained in Refs. 10-2 and 10-3, 
and basic texts on turbines and pumps are listed as Refs. 10-4 to 10-8. 

There are several design variations or geometrical arrangements for transmit- 
ting turbine power to one or more propellant pumps; some are shown schemati- 
cally in Fig. 10-4 and an initial selection has to be made. If there is a mismatch 
between the optimum pump speed and the optimum turbine speed (which is usu- 
ally higher), it may save inert mass and turbine drive gas mass to interpose a gear 
reduction between their shafts. See Fig. 6-11 and Figs. 10-4d and 10-4T For 
the last two decades designers have preferred to use direct drive, which avoids 



392 TURBOPUMPS AND THEIR GAS SUPPLIES 



the complication of a gear case but at a penalty in efficiency and the amount of 
turbine drive propellant gas required. See Figs. 6-12, 10-1, 10-2, and 10-4e. 

The key turbine parameter can be estimated because the power output of the 
turbine essentially has to equal the power demand of the pumps. If the pump is 
driven directly, that is, without a gear case, then the pump speed and the tur- 
bine speed are equal. From the properties of the turbine drive gas (temperature, 
specific heat, etc.), the strength limits of the turbine materials, and the likely 
pressure drop, it is possible to determine the basic dimensions of the blades 
[pitch line velocity, turbine nozzle outlet velocity, number of rows (stages) of 
blades, turbine type, or turbine efficiency]. The particular arrangement or geom- 
etry of the major turbopump components is related to their selection process. 
Most propellant pumps have a single-stage main impeller. For liquid hydrogen 
with its low density, a two- or three-stage pump is normally needed. Usually 
some design limit is reached which requires one or more iterations, each with a 
new changed approach or parameter. The arrangement of the major turbopump 
components (Fig. 10-4) is also influenced by the position of the bearings on the 
shaft. For example, we do not want to place a bearing in front of an impeller inlet 
because it will cause turbulence, distort the flow distribution, raise the suction 
pressure requirement, and make cavitation more likely to occur. Also, bearings 
positioned close to a turbine will experience high temperatures, which influences 
the lubrication by propellant and may demand more cooling of the bearings. 

The use of booster pumps allows lower tank pressure, and thus lower inert 
vehicle mass, and provides adequate suction pressures to the main pump inlet. 
See Ref. 10-12. Booster pumps are used in the Space Shuttle main engine and 
the Russian RD-170, as seen in Figs. 6-12 and 11-2. Some booster pumps 
have been driven by a liquid booster turbine using a small flow of high-pressure 
liquid propellant that has been tapped off the discharge side of the main pump. 
The discharged turbine liquid then mixes with the main propellant flow at the 
discharge of the booster pump. 

Mass is at a premium in all flying vehicle installations, and the feed system 
is selected to have a minimum combined mass of tubines, pumps, gas generator, 
valves, tanks, and gas generator propellants. Some of the considerations in the 
design of turbopumps are the thermal stresses, warpage due to thermal expansion 
or contraction, axial loads, adequate clearances to prevent rubbing yet minimize 
leakage, alignment of bearings, provisions for dynamic balancing of rotating 
parts, mounting on an elastic vehicle frame without inducing external forces, and 
avoiding undue pressure loads in the liquid and gas pipes. 

Vibrations of turbopumps have caused problems during development. The 
analyses of the various vibrations of components (shaft, turbine blades, liquid 
oscillations, gas flow oscillations, or bearing vibrations) are not given here. At the 
critical speed the natural structural resonance frequency of the rotating assembly 
(shaft, impellers, turbine disk, etc.) coincides with the rotation operating speed, 
a condition that must be minimized. A slight unbalance can then be amplified to 
cause significant shaft deflections, bearing failure, and other damage. The oper- 
ating speed therefore is usually lower, but occasionally higher than the critical 



10.7. APPROACH TO TURBOPUMP PRELIMINARY DESIGN 393 



speed. A large diameter stiff shaft, rigid bearings, and stiff bearing supports will 
increase this critical speed, and damping (such as the liquid lubricant him in 
the bearing) will reduce the vibration amplitude. The solving of various internal 
vibration problems, such as whirl in bearings and blade vibrations, is reported 
in Ref. 10-16 and the dynamics of the propellant flow in pumps in Ref. 10-15. 
Vibrations of adjacent components can be excited by a TP and the natural fre- 
quency of these components can be analyzed. If the operating frequency or 
the shaft speed of the rotating assembly coincides with the natural frequency, 
another engine part, such as injector domes or certain piping assemblies, then an 
undesirable resonance with excessive stresses can occur. As a result the natural 
frequencies of affected component or of the TP can be changed by design. For 
example, whenever the tip of a pump blade goes by the tongue of the pump 
volute, a pressure wave is generated in the liquid. The frequency is the product 
of the number of pump vanes and the speed of the vane tip. Vibrations external 
to the TP, such as those generated by a thrust chamber (see Chapter 9) should not 
influence the operation of a TP. This interaction, if it occurs, can be determined 
by hot fire tests of a complete engine with appropriate instruments measuring TP 
behavior. 

The bearings in most existing TPs are high precision, special metal alloy ball 
bearings (Ref. 10-13). A few are roller bearings, which have a higher radial 
load capacity. Some ball bearings can take both radial and axial loads. Early 
ball and roller bearings were limited in the loads and speeds at which they 
could operate reliably. New bearings were developed by the bearing industry. In 
some TP designs the limits of the bearing loads and speeds determined the shaft 
speed and thus the minimum size of the turbopump, rather than the cavitation 
limit of the pump. Various types of bearings, and many types of seals have 
been explored, tested in laboratory fixtures or experimental TPs. This included 
hydrostatic (precision sleeve type) bearings, foil bearings, and magnetic bearings 
(Ref. 10-13). As far as the authors know, none of these have as yet found their 
way into a production TP. The variety of static and dynamic seals is relatively 
large, and the selection is strongly influenced by the preferences of the design 
organization (Ref. 10-14). One type of bearing oscillation problem is discussed 
an Ref 10-16. 

If the turbopump is part of a reusable rocket engine, it becomes more complex. 
For example, it can include provision to allow for inspection and automatic 
condition evaluation after each mission or flight. This can include an inspection 
of bearings through access holes for horoscope instruments, checking for cracks 
in highly stressed parts (turbine blade roots or hot-gas high-pressure manifolds), 
or the measurement of shaft torques (to detect possible binding or warpage). 

The number of different materials of construction of TP appears to have 
increased. For example, for high-speed, high-load, ball bearings a new ball 
material (silicon nitride) has been successfully introduced. Relatively common 
materials, such as stainless steels, have in part been replaced by superalloys, 
such as Inconel. Powder metallurgy has found its way into impellers and turbine 
parts. Although the strength is not really better than a forged or cast material. 



394 TURBOPUMPS AND THEIR GAS SUPPLIES 



they have a smooth surface (low friction), uniform physical properties, and can 
be fabricated in complex shapes (Ref. 10-22). 

There is no warm-up time available in rocket turbines. The sudden admission 
of hot gas at full flow causes severe thermal shock and thermal distortion and 
increases the chances for rubbing between moving metal parts. The most severe 
stresses of a turbine blade often are thermal stresses; they come during the engine 
start when the leading edge is very hot but other parts of the blade are still cold. 

For low-thrust engines the shaft speeds can become very high, such as over 
100,000 rpm. Also, the turbine blade height becomes very short and friction 
losses can become prohibitive. In order to obtain a reasonable blade height we 
go to partial admission turbine designs. Here a portion of the turbine nozzles are 
effectively plugged or eliminated. 



10.8. GAS GENERATORS AND PREBURNERS 

The purpose of a gas generator or a preburner is to create the “warm” gas (usually 
between 600 and 2000°F or 315 to 1200°C) to drive the turbine of a TP. See 
Refs. 10-1 to 10-5. The selected gas temperature depends on nearly uniform 
gas composition, the engine design, the turbine blade materials, using uncooled 
hardware, and the acceptable risk of failure. In order to achieve these warm gas 
temperatures with a bipropellant, most operate at a fuel-rich mixture ratio, but 
some of the prebumers in the Soviet Union (today Russia) operate at an oxidizer- 
rich mixture usually with engines using staged combustion engine cycles. Each 
of these combustion devices consists of a combustion chamber, an injector, and 
a pipe or duct leading to the turbine. They all have separate dedicated propellant 
control valves, sometimes with some calibrated orifices, for control of the flows. 
They resemble a thrust chamber, but they do not have a single bow-tie-shaped 
supersonic nozzle. Instead the warm gas flows subsonically from the gas generator 
or the prebumer through pipes or manifolds into the nozzles of the turbine (here 
they reach high velocities) and then they flow supersonically through the row(s) 
of turbine blades, which extract the energy for driving the propellant pumps. 
Requirements for gas generators or preburners include the delivery of warm 
gas at the intended mass flows, pressure, and design temperature, an essentially 
uniform gas temperature across the flow path to the turbine and no high gas 
temperature spikes. Combustion instability problems are extremely rare. 

Gas generators are used exclusively with liquid propellant rocket engines, 
which operate on a gas generator engine cycle, and preburners are used exclu- 
sively with rocket engines, which operate on a staged combustion cycle and they 
usually operate at a higher pressure. Table 10-3 explains some of the differences 
between these two and Table 6-6 and Fig. 6-9 describe the engine cycles. 

Gas generators are shown as a component of an engine in Fig. 1-4 and 
Ref. 10-4 and prebumers in Figs. 6-1, 6-12, and 11-2. Propellants supplied 
to the gas generator or the prebumer usually are tapped off from discharges of 
the engines main pumps. When starting an engine, the turbomachinery needs to 



PROBLEMS 395 



TABLE 10-3. Comparison of Key Characteristics of Gas Generators and Prebumers 



Parameter 


Gas Generator 


Preburner 


Engine cycle 


Gas generator cycle 


Staged combustion cycle 


Chamber pressure 


Usually equal or lower than 


30-60% higher than its 




thrust chamber pressure 


thrust chamber pressure 


Mass flow as % of total 
propellant flow 


1-7 


40-85 


Cooling 


Usually uncooled 


Usually uncooled, but may 
be partially cooled 


Inert mass 


Relatively light 


Heavy 


Size 


Relatively small 


Can be large 



be brought up to speed before propellant can be supplied at pressure. This gas 
generator start has also been done with a solid propellant starting cartridge (runs 
only a few seconds), an auxiliary set of small propellant tanks pressurized by 
cold gas (also runs only for short duration), or by letting the engine “bootstrap” 
itself into a start using the modest tank pressure augmented by the liquid column 
head existing in the vehicle tanks and feed system pipe lines — usually called 
“tank head” start (requires more time to start). A discussion of engine starts 
and tank pressurization can be found in Section 6.5 and thrust chamber starts in 
Section 8.6. 

In the past monopropellant gas generators were common and both 80 or 90% 
hydrogen peroxide or pure hydrazine provided warm gas usually through catalytic 
decomposition in a bed of solid catalysts. This type has a simpler gas generator 
system (only one tank and one set of valves, instead of two), has no mixture ratio 
adjustment, and a predictable, fully reproducible uniform warm gas temperature 
without potential temperature spikes. The key disadvantages were the complica- 
tions of providing a third propellant, the potential propellant hazards, the lower 
performance, or the higher mass of propellant for making enough warm gas for 
providing the required power. 



PROBLEMS 

1. A rocket engine with two TPs delivers the fuel, namely UDMH, at a pump discharge 
pressure of 555 psia, a flow of 10.2 Ih/sec, at 3860 rpm, and a fuel temperature of 
68°F. Determine the following: 

(a) The fuel pump power for these nominal conditions. 

(b) When the fuel flow is reduced to 70% of nominal, what will be the approximate 
power level, discharge pressure, and shaft speed? Assume that the oxidizer pump is 
also reduced by 70% and so is the gas flow to the turbines, but the gas temperature 
is unchanged. 



396 TURBOPUMPS AND THEIR GAS SUPPLIES 



(c) If the allowed temperature variation of the propellants is at — 40°F and on another 
day +120°F, how will this affect the power level, shaft speed, and the discharge 
pressure of the fuel pump? 

2. What are the specihc speeds of the four SSME pumps? (See the data given in 
Table 10-1.) 



3 . Compute the turbine power output for a gas consisting of 64% by weight of H 2 O and 
36% by weight of O 2 , if the turbine inlet is at 30 atm and 658 K with the outlet at 1.4 
atm and with 1.23 kg flowing each second. The turbine efficiency is 37%. 

4 . Compare the pump discharge gauge pressures and the required pump powers for five 
different pumps using water, gasoline, alcohol, liquid oxygen, and diluted nitric acid. 
The respective specific gravities are 1.00, 0.720, 0.810, 1.14, and 1.37. Each pump 
delivers 100 gal/min, a head of 1000 ft, and arbitrarily has a pump efficiency of 84%. 
Answers: 433, 312, 350, 494, and 594 psi; 30.0, 21.6, 24.3, 34.2, and 41.1 hp. 



5. The following data are given on a liquid propellant rocket engine: 



Thrust 

Thrust chamber specific impulse 

Fuel 

Oxidizer 

Thrust chamber mixture ratio 
Turbine efficiency 
Required pump power 
Power to auxiliaries mounted on turbopump 
gear case 

Gas generator mixture ratio 
Turbine exhaust pressure 
Turbine exhaust nozzle area ratio 
Enthalpy available for conversion in turbine 
per unit of gas 

Specific heat ratio of turbine exhaust gas 



40,200 Ibf 
210.2 sec 

Gasoline (specific gravity 0.74) 
Red fuming nitric acid (sp. gr. 

1.57) 

3.25 

58% 

580 hp 
50 hp 

0.39 
37 psia 
1.4 

180 Btu/lb 
1.3 



6. Determine the engine system mixture ratio and the system specific impulse. 
Answers: 3.07 and 208 sec. 



SYMBOLS 

a gear ratio 

A area, m^(ft^) 

Cp specific heat at constant pressure, J/kg K (Btu/lb°R) 

D diameter, m (ft) 

go sea-level acceleration of gravity, 9.806 m/sec^ (32.17 ft/sec^ ) 

H head, m (ft) 

(Hs)a available pump suction head above vapor pressure, often called 

net positive suction head, m (ft) 

{Hs)r required pump suction head above vapor pressure, m (ft) 



REFERENCES 397 



L torque, Nm (ft-lbf) 

m mass flow rate, kg/sec (Ibm/sec) 

N shaft speed, rpm (rad/sec) 

Ns specific speed of pump 

P pressure, N/m^ (Ibf/in.^ ) 

P power, W (hp) 

P), power of auxiliaries, bearings, rubbing seals, friction 

Q volume flow rate, m^/sec (ft^/sec) 

S suction specific speed of pump 

T absolute temperature, K (°R) 

u impeller tip speed or mean blade speed, m/sec (ft/sec) 

V liquid fiow velocity, m/sec (ft/sec) 

Greek Letters 

A finite differential 

ri efficiency 

p density, kg/m^(lb/ft^) 

velocity correction factor 

Subscripts 

e maximum efficiency condition 

/ fuel 

o oxidizer 

p pump 

t turbine 

0 initial condition 

1 inlet 

2 outlet 



REFERENCES 

10-1. G. P. Sutton, “Turbopumps, a Historical Perspective,” AIAA Paper 2006-7531, 
July 2006. 

10-2. D. K. Huzel and D. H. Huang, “Design of Turbopump Feed Systems,” Chapter 6 
in Design of Liquid Propellant Rocket Engines, rev. ed., Vol. 147, Progress in 
Astronautics and Aeronautics , AIAA, Reston, VA, 1992. 

10-3. M. L. Strangeland, “Turbopumps for Liquid Rocket Engines,” Threshold, an Engi- 
neering Journal for Power Technology , No. 3, Rocketdyne Propulsion and Power, 
Summer 1988, pp 34-42. 

10-4. G. P. Sutton, “Turbopumps,” Chapter 4.4 and “Gas Generators, Prebumers and 
Tank Pressurization,” Chapter 4.5 in Elistory of Liquid Propellant Rocket Engines , 
AIAA, Reston, VA, 2006. 



398 



TURBOPUMPS AND THEIR GAS SUPPLIES 



10-5. Turbopump Systems for Liquid Rocket Engines, NASA Space Vehicle Design 
Monograph, NASA SP-8107, August 1974. 

10-6. Personal communications with personnel from Pratt & Whitney Rocketdyne, 
Northrop Grumman, and The Aerospace Corporation, 2006 to 2008. 

10-7. A. Minick and S. Peery, “Design and Development of an Advanced Liquid Hydro- 
gen Turbopump,” AlAA paper 98-3681, July 1998, and G. Crease, R. Lyda, 
J. Park, and A. Minick, “Design and Test Results of an Advanced Liquid Hydro- 
gen Pump,” AlAA paper 99-2190, 1999. 

10-8. 1. Kassarik, W. C. Krutzsch, W. H. Frazer, and J. P. Messina (Eds.), Pump 
Handbook, McGraw-Hill Book Company, New Tork, 1976 (water hammer and 
pumps). 

10-9. C. E. Brennan, Hydrodynamics of Pumps, Concepts ETl, Inc. and Oxford Uni- 
versity Press, 1994. 

10-10. Liquid Rocket Engines Turbopump Shafts and Couplings , NASA Space Vehicle 
Design Monograph, NASA SP-8101, Sept. 1972. 

10-11. Liquid Rocket Engines Turbopumps Gears, NASA Space Vehicle Design Mono- 
graph, NASA SP-8100, March 1974. 

10-12. Y. V. Demyanenko, A. I. Dimitrenko, and 1. 1. Kalatin, “Experience of Developing 
Propulsion Rocket Engine Feed Systems Using Boost Turbopump Units,” AlAA 
Paper 2003-5072, 2003. 

10-13. Liquid Rocket Engine Turbopump Bearings , NASA Space Vehicle Design Mono- 
graph, NASA SP-8048, March 1971. 

10-14. Liquid Rocket Engine Turbopump Rotating Shaft Seals . NASA SP-8121, February 
1978. 

10-15. J. Kurokawa, K. Kamijo, and T. Shimura, “Axial Thrust Analysis on LOX-Pump,” 
AlAA Paper 91-2410, June 1991. 

10-16. M. C. Ek, “Solving Synchronous Whirl in High Pressure Turbine Machinery of 
the Space Shuttle Main Engine,” Journal of Spacecraft and Rockets, Vol. 17, 
No. 3, May-June 1980, pp. 208-218. 

10-17. R. S. Ruggeri and R. D. Moore, Method for Prediction of Pump Cavitation Per- 
formance for Various Liquids, Liquid Temperatures, and Rotating Speeds, NASA 
TN D5292, June 1969. 

10-18. Liquid Rocket Engine Turbopump Inducers, NASA Space Vehicle Design Mono- 
graph, NASA SP-8052, May 1971. 

10-19. T. Shimura and K. Kamijo, “Dynamic Response of the LE-5 Rocket Engine 
Oxygen Pump,” Journal of Spacecraft and Rockets, Vol. 22, No. 2, March- April, 
1985. 

10-20. Liquid Rocket Engine Turbines, NASA Space Vehicle Design Criteria Monograph, 
NASA SP-8110, January 1974. 

10-21. S. Andersson and S. Trollheden, “Aerodynamic Design and Development of a 
Two-Stage Supersonic Turbine for Rocket Engines,” AlAA Paper 99-2192, 1999. 

10-22. D. Guichard and A. DuTetre, “Powder Metallurgy Applied to Impellers of Vinci 
Turbopump,” International Symposium for Space Transportation of the XXI Cen- 
tury, in CD ROM of the symposium. May 2003. 



CHAPTER 11 



ENGINE SYSTEMS, CONTROLS, 
AND INTEGRATION 



This chapter discusses the propellant budget, performance of complete or mul- 
tiple rocket propulsion systems, the design of liquid propellant rocket engines 
with pressurized or turbopump feed systems, engine controls, engine calibration, 
system integration, and system optimization. Some of the content also applies to 
solid propellant motors or hybrid propulsion systems. 



11.1. PROPELLANT BUDGET 

In all liquid propellant rocket engines the amount of propellant put into the 
vehicle tanks is always a little greater than the nominal amount of propellant 
needed to accomplish the intended mission. The extra propellant is needed for 
uses other than providing thrust (e.g., residual propellant or valve actuation), to 
compensate for changes from engine to engine, such as dimensional tolerances 
causing slight changes in flow, for uncertainties of the construction of the engine 
and minor changes in the flight plan. A propellant budget is the sum of all the 
propellant utilization categories and losses in an engine; 11 are listed below. See 
Ref. 11-1. The budget helps to determine how much propellant has to be loaded. 
It is the aim to minimize this amount of propellant. 

1. Enough propellant has to be available for achieving the required vehicle 
velocity increase and/or the nominal attitude control maneuvers of the par- 
ticular application and the particular flight vehicle or stage. The nominal 
velocity increment is usually defined by systems analysis and mission opti- 
mization using an iterative calculation based on Eq. 4-19 or 4-35. If there 
are alternative flight paths or missions for the same vehicle, the mission 



399 



400 ENGINE SYSTEMS, CONTROLS, AND INTEGRATION 

with an unfavorable flight path, such as higher drag or different orbit, and 
the highest total impulse should be selected. This mission-required propel- 
lant is usually the largest portion of the total propellants loaded into the 
vehicle tanks. 

2. In a turbopump system using a gas generator cycle, a small portion of the 
overall propellant is burned in a separate gas generator. It has a lower 
flame temperature than the thrust chamber gas and operates at a different 
mixture ratio; this causes a slight change in the overall mixture ratio of 
propellants flowing from the tanks, as shown by Eqs. 11-3 and 11-5. 

3. In a rocket propulsion system with a thrust vector control (TVC) system, 
such as a swiveling thrust chamber or nozzle, the thrust vector will be 
rotated by a few degrees. Thrust vector control systems are described in 
Chapter 18. There is a slight decrease in the axial thrust and that reduces 
the vehicle velocity increment in item 1. The extra propellant needed to 
compensate for the small velocity reduction can be determined from the 
mission requirements and TVC duty cycle. It could be between 0.1 and 
4% of the total propellant depending on the average angle position of this 
thrust chamber. 

4. In some engines a small portion of cryogenic propellants is heated, vapor- 
ized, and the gas used to pressurize cryogenic propellant tanks. A heat 
exchanger is used to heat liquid oxygen from the pump discharge and pres- 
surize the oxygen tank, as shown schematically in Fig. 1-4. This method 
is used in the hydrogen and oxygen tanks of the Space Shuttle external 
tank (see Ref. 6-9). 

5. Auxiliary rocket engines that provide for trajectory corrections, station 
keeping, maneuvers , or attitude control usually have a series of small 
restartable thrusters (see Chapter 4). The propellants for these auxiliary 
thrusters have to be included in the propellant budget if they are supplied 
from the same feed system and tanks as the larger rocket engine. Depend- 
ing on the mission, the duty cycle, and the propulsion system concept, 
this auxiliary propulsion system can consume a signiflcant portion of the 
available propellants. 

6. The residual propellant that clings to tank walls or remains trapped in 
valves, pipes, injector passages, or cooling passages is unavailable for pro- 
ducing thrust. It is typically 0.5 to 2% of the total propellant load. All 
unused residual propellant increases the flnal vehicle mass at thrust termi- 
nation and reduces the flnal vehicle velocity slightly. 

7. A loading uncertainty exists due to variations in tank volume or changes 
in propellant density or liquid level in the tank. This is typically 0.25 to 
0.75% of the total propellant. It depends, in part, on the accuracy of the 
method of measuring the propellant mass during loading (weighing the 
vehicle, flow meters, level gauges, etc.). 

8. The off-nominal rocket performance is due to variations in the manufacture 
of hardware from one engine to another (such as slightly different pressure 



1 1 .2. PERFORMANCE OF COMPLETE OR MULTIPLE ROCKET PROPULSION SYSTEMS 401 



losses in a cooling jacket, in injectors and valves, or somewhat different 
pump characteristics); these cause slight changes in combustion behavior, 
mixture ratio, or specific impulse. If there are slight variations in mixture 
ratio, one of the two liquid propellants will be consumed fully and an 
unusable residue will remain in the other propellant’s tank. If a minimum 
total impulse requirement has to be met, extra propellant has to be tanked to 
allow for these mixture ratio variations. This can amount to up to perhaps 
2.0% for each of the propellants. 

9. Operational factors can result in additional propellant requirements, such 
as filling more propellant than needed into a tank or incorrectly adjusting 
regulators or control valves. It can also include the effect of changes in 
flight acceleration from the nominal value. For an engine that has been 
carefully calibrated and tested, this factor can be small, usually between 
0.1 and 1.0%. 

10. When using cryogenic propellants, an allowance for evaporation and cool- 
ing down has to be included. It is the mass of extra propellant that is 
allowed to evaporate (and be vented overboard while the vehicle is waiting 
to be launched) or that is fed through the engine to cool it down, before the 
remaining propellant in the tank becomes less than the minimum needed 
for the flight mission. Its quantity depends on the amount of time between 
topping off (partial refilling) of the tank and the engine start. 

1 1 . Finally, an overall contingency or ignorance factor is needed to allow for 
unforeseen propellant needs or inadequate or uncertain estimates of any 
of the items above. This can also include allowances for vehicle drag 
uncertainties, variations in the guidance and control system, wind, or leaks. 

Only some of the items above provide axial thrust (items 1, 2, and sometimes 
also 3 and 5), but all the items need to be considered in determining the total 
propellant mass and tank volume. 

The example shown in Table 11-1 is for a spacecraft pressure-fed engine 
system, where the majority of the propellant is consumed in a larger axial thrust 
chamber, and the second largest amount of propellant is fed to a set of several 
small thrusters used for extensive attitude control maneuvers. There will be flights 
where the mission may be more demanding or the engine performance may be 
slightly lower, and extra propellant will be needed to accomplish the mission. 
Conversely if the engine performance is actually slightly better than nominal 
or where the mission can be accomplished with less total impulse (operate with 
fewer or lower orbits), then the engine will consume less than the nominal amount 
of propellant. 

11.2. PERFORMANCE OF COMPLETE OR MULTIPLE ROCKET 
PROPULSION SYSTEMS 

The simplified relations that follow give the basic method for determining the 
overall specihc impulse, the total propellant flow, and the overall mixture ratio 



402 ENGINE SYSTEMS, CONTROLS, AND INTEGRATION 



TABLE 11-1. Example of a Propellant Budget for a Spacecraft Propulsion System 
with a Pressurized Monopropellant Feed System 

Budget Element Typical Value 



1 . Main thrust chamber (increasing the velocity 
of stage or vehicle) 

2. Flight control function (for reaction control 
thrusters and flight stability) 

3. Residual propellant (trapped in valves, lines, 
tanks, etc.) 

4. Loading uncertainty 

5. Allowance for off-nominal performance 

6. Allowance for off-nominal operations 

7. Mission margin (reserve for first two items 
above) 

8. Contingency 



85-95% (determined from mission 
analysis and system engineering) 
5-10% (determined by control 
requirements) 

0.5-2% of total load 

0.5% of total load 
0.5 -1.0% of total load 
0.25-1.0% of total load 
3-5% of items 1 and 2 

1-5% of total load 



Source: Adapted from data supplied by predecessor of Propulsion Products Center, Northrop Grum- 
man Corporation 



as a function of the corresponding component performance terms for complete 
rocket engine systems. This applies to engine systems consisting of one or more 
thrust chambers, auxiliaries, gas generators, turbines, and evaporative propellant 
pressurization systems all operating at the same time. 

Refer to Eqs. 2-5 and 6-1 for the specific impulse /j, propellant flow rate w 
or m and mixture ratio r. The overall thrust Fga is the sum of all the thrusts from 
thrust chambers and turbine exhausts and the overall flow tit is the sum of their 
flows as shown in Eqs. 2-24 and 2-25. The subscripts oa, o, and / designate 
the overall engine system, the oxidizer, and the fuel, respectively. Then repeating 
these equations 

(/)oa — 



W„ 



T.F 


T.F 

goT,Fi 


(11-1) 




or fkoa = m 


(11-2) 




_J2Fio 


(11-3) 


Ew'f 





These same equations should be used for determining the overall performance 
when more than one rocket engine is contained in a vehicle propulsion system and 
they are operating simultaneously. They also apply to multiple solid propellant 
rocket motors and combinations of liquid propellant rocket engines and solid 
propellant rocket booster motors, as in the Space Shuttle (see Eig. 1-14). All the 
nozzles point in the same direction in Eqs. 2-24, 11-1, and 11-4. 

Example 11-1. For an engine system (LOX/kerosene) with a gas generator similar to 
the one shown in Fig. 1-4, determine a set of equations that will express (1) the overall 



11.3. ENGINE DESIGN 403 



engine performance and (2) the overall mixture ratio of the propellant flows from the tanks. 
Let the following subscripts be used: c, thrust chamber; gg, gas generator; and tp, tank 
pressurization. For a nominal burning time f, a 1% residual propellant, and a 6% overall 
reserve factor, give a formula for the amount of fuel and oxidizer propellant required 
with constant propellant flow. Ignore stop and start transients, thrust vector control, and 
evaporation losses. 



SOLUTION. Only the oxidizer tank is pressurized by vaporized propellant. Although 
this pressurizing propellant must be considered in determining the overall mixture ratio, 
it should not be considered in determining the overall specific impulse since it stays with 
the vehicle and is not exhausted overboard. 



(f.v)oa 



^ oa 



Fc + Fgg 

{me + mgg)go 

{FIq^C “t“ (hlo)gg “t“ {^o)tp 

(mf)c + (mf)gg 



(11-4) 

(11-5) 



nif = [{mf)c + (m/)gg]?(1.00 + 0.01 + 0.06) 

nio = [{mo)c + (ffio)gg + («L )?;,]? (1-00 + 0.01 + 0.06) 

For this gas generator cycle the engine mixture ratio or Vga is different from the 
thrust chamber mixture ratio rc = {m„)c/{mf)c. Similarly, the overall engine specific 
impulse is slightly lower than the thrust chamber specific impulse L = Fe/ihe. However, 
for an expander cycle or a staged combustion cycle these two mixture ratios and two 
specific impulses are the same, provided that there are no gasified propellanf used for 
tank pressurization. Engine cycles are explained in Section 6.6. 

The overall engine specific impulse is influenced by the propellants, the nozzle 
area ratio, and the chamber pressure, and to a lesser extent by the engine cycle, 
and the mixture ratio. Table 11-2 describes 10 different rocket engines using 
liquid oxygen and liquid hydrogen propellants designed by different companies 
in different countries, and shows the sensitivity of the specific impulse to these 
parameters. References 11-2 to 11-4 give additional data on several of these 
engines. 



11.3. ENGINE DESIGN 

The approach, methods, and resources used for rocket engine preliminary design 
and final design are usually different for each design organization and for each 
major type of engine. They also differ by the degree of novelty. 

1. A totally new engine with new major components and some novel design 
concepts will result in an optimum engine design for a given application, 
but it is usually the most expensive and longest development approach. 
One of the major development costs is usually in sufficient testing of 



TABLE 11-2. Comparison of Rocket Engines Using Liquid Oxygen and Liquid Hydrogen Propellants' 



CJ 

.3 




bX) 






















c/3 

c/3 


4^ 


o 


o 


in 


o 


in 


in 


CN 


in 


NO 


O 


'5b 


3 




o 


o 


in 


CN 


00 


in 


m 


i> 


m 


»n 


3 

PJ 




£:• 

■o 




00 

NO 


CN 


r- 


in 






CN 


CN 


in 



n « 

>§ < ai. 



00 



*'1 

CN 



O 












in 

cn 



o 



o 

00 



0 ) 

S-I 

D 

§ 



o 






o 



53 



o 


o’ 




'S 


l-H 

3 

c/3 


c/3 

3- 


3 

4= 


C/3 

O 

1-1 




u 


CP 


40 






o 



o 

in 



o 



in 

m 

in 



o o q o 
in in »n 



O 


CS 


o 


CN 


ON 

NO 


CN 


CNJ 


CN 


»n 


in 

i>; 


^ CM 
»n ^ 


_ 00 
O CM 


- 


r- 


CM 




l> 




CN 


l> 




NO 


NO 


CN 


c4 




\o 


00 


NO 


cn 




m 


in 










m 


»n 


m 


s 5 


s ^ 


m 


cn 


in 



^ o ^ 


in 






q 




CM 




in 




'^'3 3 


CM 


o 


CM 


in 


m 






NO 


o 


0 3 3 


in 




in 




m 






NO 


CM 


Oh Oh 3 

C/5 S d 
















''t 





^23 



o 




o 




o 






o 




o 




















^ 


»n 


cn 


o 


in 


^ O 


o 




''t 




o 




o 




o 




o 


in 


o 


q 


q^ 


* 


<N § 


q 


CM 


00 




o 




o 


(3 


in 


'55l- 


o 




o 


o' 


m 


in 


CM 


O 


cm" 




CM* 




rn 


»n 




r-; 




o 


00 


q 


ON 






1— 


r- ^ 




1-H 


in 


NO 




r- 


no" 






'55l- 


o" 


r- 


o" 






C-- 




CM 


CM 




CM 












CM 











4= 



C 3 3 
O- 4 = 
00 C/5 



3 

cr 

D . 



<D 

-Q 



•< (M •< 



cu 

bO 

3 

c/3 


c/3 

3 

.2 


c/3 

Vh 

O 


c/3 

3 

O 


O 

> 




‘C 


3- 


£ 


o 






C4 


3 


40 


m 


> 


3 


c/3 


3 



M (D 

OX) 

3 

O "S 
t— ) m 



(U 

Mb 

W § 

^ o 

a o ; 

§ • 



Q S 



W) 

C3 

PJ u 



3 
3 

a 



CD 

C 

- "2 
3 Q 
O 4^ 
'z: o 
cfl O 

== c2 

40 ^ 

3 >> 

o S 

-3 2 

bJ) > 
3 






PJ 

S 

00 

00 



00 

ON 

ON 



0^ 



2 

3 

Vh 

OJ 

3 

O 

bX) 

on 

3 

biD 

oo" 

q 



8 £ 

D *0 

S, 

"2 X5 



2^ D 

O "O 
O 3 
V 3 
^ Oh 

g w 
> »o 



3 

40 

E 

' o ^ 

' C> (N 

' T5 
; ^ ON 

^ 00 
. 3 ^ 



c« 



3 
3 

C-’' ^ 
PP ^ 



Oh 

pq 

c/5 

Vh 

o 

3 

1 -H 

O 

3 
<1^ ^ 
OX Co 

M ON 
3 ON 
bfl ^ 

e ^ 
3 O 



Ph 

w 

c/5 

sP' 

O 

3 

^ s 

C ^ 

g 00 

bX) 2 

Cfl ^ 

M 8 



J 



X 



u 

CU 



T3 

3 

3 

C 4 

X 

o 



in 

NO 

ON 



00 

ON 

3 

O 

1 -H 



3 

.s 

2 ? 
^ PU 
>H 



3 

3 

S 

u 

(D 

.3 

'3 ; 2 T 

u 

Vh on 

ci”; 

«i E 
5 ^ O 

Oh ^ 



404 



Additional information on most of these engines is in this book. Use the index to find it. 



11.3. ENGINE DESIGN 405 



components and several engines (under various environmental and perfor- 
mance limit conditions), in order to establish credible reliability data with 
enough conhdence to allow the initial flights and initial production. Since 
the state of the art is relatively mature today, the design and development 
of a truly novel engine does not happen very often. 

2. New engine using some major components or somewhat modified key com- 
ponents from proven existing engines. This is a common approach today. 
The design of such an engine requires working within the capability and 
limits of existing or slightly modified components. It requires less testing 
for proving reliability. 

3. Uprated or improved version of an existing, proven engine. This approach 
is quite similar to the second. It is needed when an installed engine for 
a given mission requires more payload (which really means higher thrust) 
and/or longer burning duration (more total impulse). Uprating often means 
more propellant (larger tanks), higher propellant flows and higher chamber 
and feed pressures, and more feed system power. The engine usually has 
an increased inert engine mass (thicker walls). 

In a simplified way, we describe here a typical process for designing an engine. 
Chapter 19 and Refs. 11-5 and 11-6 describe this process and the selection of the 
propulsion system from a different point of view. At first the basic function and 
requirements of the new engine must be established. These engine requirements 
are derived from the vehicle mission and vehicle requirements, usually deter- 
mined by the customer and/or the vehicle designers, often in cooperation with 
one or more engine designers. The engine requirements can include key param- 
eters such as thrust level, the desired thrust-time variation, restart or pulsing, 
altitude flight profile, duty cycle, maximum accelerations, engine locations within 
the vehicle, and limitations or restraints on cost, engine envelope, test location, or 
schedule. It also includes some of the factors listed later in Table 19-5. If an exist- 
ing proven engine can be adapted to these requirements, the subsequent design 
process will be simpler and quite different than the design of a truly new engine. 

Usually some early tentative decisions about the engine are made, such as the 
selection of the propellants, their mixture ratio, or the cooling approach for the 
hot components. They are based on mission requirements, customer preferences, 
past experiences, some analysis, and the judgment of the key decision makers. 
After some studies additional selection decisions can also be made, such as having 
one, two, or more thrust chambers fed from the same feed system, redundancy 
of auxiliary thrusters, or type of ignition system. 

A systematic approach, a good set of analyses, the use of system engineering, 
good coordination with the customers, key vendors, and vehicle designers — all 
are needed for a good preliminary and final design. Before a meaningful proposal 
for an engine can be prepared, a preliminary design has to be completed. See 
Refs. 11-5 to 11-7. One of the early design decisions is the choice of feed 
system: pressurized gas feed or pump feed. The next two paragraphs give some 
guidelines. 



406 ENGINE SYSTEMS, CONTROLS, AND INTEGRATION 



A pressurized feed system (see Fig. 1-3) usually gives better vehicle perfor- 
mance for low values of total impulse (thrust less than about 4.5 kN or 1000 Ibf 
with up to perhaps 2 min duration). A pump feed system gives better vehicle 
performance for high thrust (say above 50,000 Ibf or approximately 222 kN) and 
a long cumulative duration — more than a couple of minutes. For intermediate 
values of total impulse the choice can go either way, and it is not easy to make 
the decision only on the basis of total impulse. If the chamber pressure of a 
pressurized feed system is relatively high (say about 2.4 to 3.5 MPa or about 350 
to 500 psia and occasionally more), then the inert weight of the thrust chamber 
will be high, but the thmst chamber will be small and can usually ht into an 
engine compartment. The vehicle propellant tanks and pressurizing gas tank will 
be at relatively high pressure and will be heavy. For a relatively low chamber 
pressure (0.689 to 1.379 MPa or 100 to 200 psia) the vehicle tank pressure will 
be lower and the tank walls thinner, but the thrust chamber will be very large 
in size and often will exceed the limits of the engine compartment, unless it 
has a low nozzle area ratio, which implies lower performance. Even in some 
cases of low chamber pressure, a pressurized feed system can be heavier than 
a comparable engine with a pump feed system. Pressurized feed systems have 
been relatively simple, very reliable, and they allow fast starts and fast restarts. 
With restarts the cooling of the thrust chamber can become a problem. Because 
of the proven reliability of pressure feed system, NASA has at times conser- 
vatively selected pressurized feed systems for certain space applications, such 
as the Apollo service module engine (21,900 Ibf thrust), even though there is a 
major weight penalty and a somewhat inferior vehicle performance compared to 
a pump-fed system of equal total impulse. Also a decision has to be made on 
using either a pressurized system with a gas pressure regulator or alternatively a 
blow-down system. See Table 6-3 and Section 6.4. 

For a turbopump-fed liquid propellant rocket engine (see Fig. 1 -4) the overall 
inert weight of propellant tanks and engine will be considerably lighter and 
usually the vehicle performance will be somewhat better. It usually operates 
at high chamber pressures (3.5 to 24.1 MPa or about 500 to 3500 psia), the 
thrust chamber is not normally protrading from the vehicle. The smaller and 
shorter thrust chamber often allows a shortening of the vehicle with a savings 
in vehicle structure. This further improves the vehicle’s performance modestly 
and the higher specific impulse will slightly reduce the amount of propellant 
needed for the mission. Compared to an engine with a pressurized feed system, 
the savings in inert mass (thin vehicle tank walls) and less propellant will allow 
a smaller, lighter, and probably lower cost vehicle with a somewhat superior 
performance. The engine with a TP is more complex with more parts and the 
engine will generally be heavier; however, the vehicles propellant and gas tanks 
will be much lighter and they will more than compensate for the heavier engine. 
It will take more tests and more effort to prove high reliability in an engine with 
a pump feed system. At the higher chamber pressures the heat transfer will be 
higher and cooling can become a problem, but heat transfer has been solved in 
other earlier high-pressure rocket engines. Also restart will be more complex. One 



11.3. ENGINE DESIGN 407 



of several engine cycles has to be selected. High reliability has been achieved in 
many turbopump-fed large rocket engines. 

Trade-off studies between several options are appropriate at this time. With a 
modified existing engine these parameters are well established and require fewer 
trade-off studies or analyses. Initial analyses of the pressure balances, power 
distribution between pumps and turbines, gas generator flow, propellant flows 
and reserves, or the maximum cooling capacity are appropriate. Sketches and 
preliminary estimates of inert mass of key components need to be made, such as 
tanks, thrust chambers, turbopumps, feed and pressurization systems, thrust vector 
control, or support structure. Alternate arrangements of components (layouts) are 
usually examined, often to get the most compact configuration or control of the 
travel of the center of gravity. An initial evaluation of combustion stability, stress 
analysis of critical components, water hammer, engine performance at some off- 
design conditions, safety features, testing requirements, cost, and schedule are 
often performed at this time. Participation of appropriate experts from the field of 
manufacturing, field service, materials, stress analysis, or safety can be critical for 
selecting